Ray optics: Prism and Optical Instruments

1. Refraction through a prism is show in figure. After suffering refraction at two faces of a prism, the emergent ray is always found to bend towards the base of the prism. It is observed, that \[\angle A = \angle r_1 + \angle r_2\]and \[\angle A + \angle \delta = \angle i + \angle e\]Angle between the incident ray and the emergent ray i.e., $ \angle \delta $ is known as the angle of deviation. Its value depends upon the angle of incidence, refractive index of prism material and the angle of prism.

2. When refracted ray passes symmetrically through a prism i.e., when $ r_1 $ = $ r_2 $ and i = e, the light rays undergoes minimum deviation Dm and in such an eventuality, \[n_{21} = \frac{sin\frac{A+D_m}{2}}{sin{\frac{A}{2}}}\]where $ n_{21} $ is the refractive index of prism material with respect to the medium outside.

3. For a prism of small angle (i.e., if  $ \angle  A $ is small enough), the angle of deviation is given by \[\delta , D = (n_{21}-1)A\]

4. Dispersion is the phenomenon of splitting of light into its component colours (or wavelengths) on passing through a dispersive medium. The pattern of colour components of light is called its spectrum. For sunlight, the spectrum consists of seven constituent colours given by the acronym VIBGYOR. In white light spectrum the violet ray is deviated the most and the red ray is deviated the most and the red ray is deviated the least.

5. Cause of dispersion in variation of refractive index with wavelength of light. In fact, \[n = A + \frac{B}{\lambda ^2}\]where A and B are two constants are a given material. As a result, the refractive index of prism and consequently the angle of deviation is maximum for violet colour ray and least for red colour ray. It results in dispersion.

6. Angular dispersion produced by a prism for white light is difference in the angles of deviation of two extreme colours i.e., violet and red colours. Mathematically, Angular dispersion = $ \delta_v - \delta_r = (n_v - n_r)A $.

7. The light, while passing through earth’s atmosphere, gets scattered by the atmospheric particles. According to Rayleigh’s law of scattering, for scattering from tiny scattering objects e.g., air molecules the intensity of the light corresponding to a wavelength in the scattered light varies inversely as the fourth power of the wavelength. Mathematically, Amount of scattering $ \propto \frac{1}{\lambda^4} $

8. Blue colour of sky, blue colour of ocean water, reddish appearance of Sun at sunrise or sunset are some common phenomenon based on Rayleigh’s scattering. Due to this very reason, red light is used in danger signals.

9. Rainbow is an example of dispersion of light, caused by tiny water droplets hanging in the atmosphere after the rains.

10. The human eye is one of the most valuable and sensitive sense organ, the human being have. Our eyes have a lens system which focus the light rays coming from an object on the retina. Retina contains rods and cone which sense light intensity and colour respectively. Retina transmits electrical signals via the optic nerve to the brain, which analyses the information received and perceives the object.

11. The eyelens has the power of accommodation t adjust it focal length so as to focus objects situated at different distance form eye at the retina.

12. The least distance of distinct vision or near point of an eye is the minimum distance from the eye at which object can be seen distinctly. For a young adult with normal vision near point is at 25 cm.

13. The farthest point up to which an eye can see objects clearly is called the far point of eye. For a normal vision, the far point of eye lies at infinity. In this situation, our eye is least strained.

14. There are four common defects of vision. These are (i) myopia or short-sightedness (ii) hypermetropia or long-sightedness (iii) presbyopia and (iv) astigmatism.

15. A myopic eye can see near objects clearly but cannot see far off objects clearly i.e., the far point of defective eye is not at infinity but has shifted nearer to the eye. This defect may arise either due to (a) excessive curvature of the cornea, or (b) elongation of eyeball. The defect can be corrected by use of a concave (diverging) lens of appropriate power.

16. In hypermetropia, a person can see distance objects clearly but cannot see nearby objects distinctly i.e., for defective eye the near point has shifted away from the eye. This defect arises either due to less curvature of cornea or contraction of the eyeball. The defect can be corrected by use of convex (converging) lens of appropriate power. With increase in age the ciliary muscles gradually weaken and power of accommodation of eye decreases. It is called presbyopia. It can be corrected by using a converging lens for reading .

17. In astigmatism, a person cannot focus simultaneously on both horizontal and vertical lines. It arises when the cornea is not spherical in shape. The problem can be rectified by using cylindrical lens of desired radius of curvature with an appropriately directed axis.

18. A microscope is used for observing magnified images of nearby tiny objects. A simple magnifier or microscope is a convex lens of small focal length held near the object such that $ u \leq  f $.

19. In a simple microscope if image is formed at near point, the angular magnification of image is $ m = (1 + \frac{D}{f} ) $. However, if image is formed at infinity then magnification $ m = \frac{D}{f} $.

20. A compound microscope consists of two convex lenses, an objective lens of very small focal length ($ f_0 $) and small aperture and an eye lens of small focal length ($ f_e $) and slightly greater aperture, placed coaxially at a suitable fixed distance of distinct vision (D = 25 cm) from the eye and is virtual, inverted and highly magnified.

21. The angular magnification of a microscope is defined as the ratio of the angle subtended by the final image at the eye to the angle subtended by the object at the eye when seen directly. Angular magnification of a compound microscope is given by : 

(a) If final image is formed at near point of eye, then \[m = m_0 \times m_e = -\frac{v_0}{u_0}(1+\frac{D}{f_e})=-\frac{L}{f_0}(1+\frac{D}{f_e})\]

(b) If final image in a microscope is formed at infinity, then \[m =-\frac{L}{f_0}\frac{D}{f_e}\]

22. The resolving power of a compound microscope is its ability to show as distinct (separate), the images of two point objects lying close to each other. The limit of resolution of a microscope is measured by the minimum distance d between two point objects, whose images in microscope are seen as just separate. It is found that \[d = \frac{1.22 \lambda}{2nsin\alpha} = \frac{0.16 \lambda}{N.A.}\]where n = refractive index of medium between the object and the objective lens, $ 2\alpha $ = angle subtended by the diameter of objective lens at the focus point and N.A. = $ n sin \alpha $ = numerical aperture of objective. Resolving power of a microscope is the reciprocal of its limit of resolution. For higher resolving power the numerical aperture the numerical aperture of objective lens of microscope should be large and wavelength of light used should be as small as possible.

23. An astronomical telescope is used to form magnified and distinct images of heavenly bodies like planets stars, moons, galaxies etc. A refracting type astronomical telescope consists of a convex objective lens of large focal length and large aperture and another convex eyepiece lens of small total length and small aperture. Final image formed is inverted, magnified and at infinitly in normal adjustment.

24. The angular magnification of a telescope is defined as the ratio of the angle subtended at the eye by the final image to the angle subtended at the eye by the object directly. It is found that in normal adjustment \[m = -\frac{f_0}{f_e}\]and length of telescope tube $ L = f_0 + f_e $. 

25. In a reflecting type telescope we use a concave mirror (generally parabolic) of large aperture and large focal length as the objective and a convex lens of small focal length and aperture as the aberrations, are cheap, easy to construct and handle.

26. The limit of resolution of a telescope is measured by the angle ($ \Delta \theta $) subtended at its objective, by those two distant objects whose images are just seen separate through the telescope.

Resolving power of telescope = $ \frac{1}{\Delta \theta}= \frac{A}{1.22 \lambda}    $, where A is the aperture size of the telescope objective.

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Ray Optics: Reflection and Refraction

1. Light is that form of energy which causes sensation of sight of your eyes. In fact, light is a part of electromagnetic radiation spectrum having its wavelength ranging from about 400 nm to 700nm.

2. In vacuum light of any wavelength (or any frequency) travels with a speed c = $ 2.99792458 \times 10^8 $ m/s but for ordinary calculations this value may be considered as c = $ 3 \times 10^8 $ m/s. The speed of light in vacuum is the highest speed attainable in nature. Moreover, speed of light in vacuum is independent of the relative motion between the source and the observer. No physical signal or message can travel with a speed greater than c.

3. As the wavelength of light is very small compared to the size of ordinary objects, a light wave can be considered to travel from one point to another point along a straight line path. Such a path is called a ray of light of right and a bundle of rays constitutes a beam of light.

4. In reflection of light, the light rays return to the same medium after striking the surface of another medium (say a mirror). The wavelength and the speed of light remains the same.

5. There are two basic laws of reflection, which are followed for every short of reflection. According to these

(i)  The incident ray, reflected ray and the normal at  the point of incidence all lie in the same plane.

(ii)  Angle of incidence (i) = angle of reflection (r).

6. For reflection from a plane mirror, the image formed is always erect, virtual and laterally inverted. The image is of exactly the same size as the object and the image is formed as such behind the mirror as the object is in front of it.

7. For a spherical mirror in the mid-point of reflecting surface is called its pole. The line passing through pole and the center of curvature of mirror is called its principle axis.

8. The principle focus of a spherical mirror is a point on its principle axis, where a beam of light incident parallel to principle axis of mirror, after reflection, actually converges to (in case of a concave mirror) or appears to diverge from (in case of a convex mirror). Distance of principle focus from pole is called the local length of given mirror.

9. Focal length of a spherical mirror is half of its radius of curvature i.e., f = R/2.

10. As per Cartesian sign convention system for mirrors, the light ray is taken to travel from left to right. All distances are measured from the pole as origin. The distance measured in the same direction as the incident light are taken as positive and those measured in the opposite direction are taken as negative. Thus, the distances to the right of pole will be + ve but distances to the left of pole will be – ve. Again distances above the principle axis are taken as + ve but distances below it – ve.

11. A concave mirror may form either a real or a virtual image depending upon the position of the object relative to the mirror. A convex mirror forms only virtual images.

12. If an object is placed at a distance u from the pole of a mirror of focal length f and its image is formed at a distance v from the pole, then according to mirror formula, we have \[\frac{1}{u} + \frac{1}{v}=\frac{1}{f}=\frac{2}{R}\].

13. If a thin linear object of height h is situated normally on principle axis of mirror at a distance u and its image of height h’ is formed at a distance v from the pole, then the linear magnification m is defined as  \[M = \frac{h'}{h} = -\frac{v}{u} = \frac{f}{f-u} = \frac{f-v}{f}\]-ve magnification means inverted image and +ve magnification means erect image.

14. When a light ray travels obliquely from one transparent medium to another, it changes the direction of its path at the interface of the two media. This is called “refraction” of light.

15. There are two laws of refraction, which are as follows:

(i)  The incident ray, the refracted ray and the normal to the interface at the point of incidence, all lie in the same plane.

(ii)    The ratio of the sine of the angle of incidence in 1^st medium t sine the angle of refraction in second medium is a constant, knows as the refractive index of 2^nd medium with respect to the 1^st medium. Mathematically,  \[\frac{sin (i)}{sin (r)} = n_{21}\]Second law of refraction is known as Snell’s law.

16. Value of refractive index depends upon the pair of media and the wavelength of light but is independent of the angle of incidence.

17. When a light ray obliquely enters from an optically rarer medium to an optically denser medium, the light ray bends towards the normal. However, if a light ray travels from denser to rarer medium, it bends away from the normal.

18. Absolute refractive index of a transparent medium is defined as the ratio of the speed of light vacuum (c) to the speed of light in given medium (v) i.e., $ n = \frac{c}{v}$. It can be show that $ n_{21} = \frac{n_2}{n_1} = \frac{v_1}{ v_2} = \frac{\lambda_1}{\lambda_2}$. It is found that $n_{12} = \frac{1}{n_{21}} $.

19. When a light ray passes through a parallel sided slab of a transparent medium, the final emergent ray is parallel to the incident ray, but is laterally displaced by a distance d given by \[d = t \frac{sin (i – r)}{cos r}\]The value of lateral shift depends upon (a) thickness (t) of the transparent slab, (b) angle of incidence (i) and (c) refractive index of the material of slab.

20. When an object situated in medium number 2 is viewed from medium number 1, the apparent depth (height) of object appears to be different from its real depth 9height) and these are co-related as: \[\frac{d_{Real }}{ d_{Apparent}} = n_{12} = \frac{1}{n_{21}}\]

21. If an object situated in an optically denser medium is viewed by an observer situated in optically rarer medium, the apparent height is less than its real height. However, if an object situated in rarer medium is viewed by an observer situated in denser medium, then the apparent height is found to be more than its real height.

22. On account of atmospheric refraction the Sun is visible about 2 minutes before the actual sunrise and for 2 minutes even after the actual sunset. Thus, Sun also appear to be of oval shape at the time of sunrise or sunset on account of atmospheric refraction.

23. For a pair of media in contact, circuital angle is the angle of incidence in the denser medium corresponding to which angle of refraction in the rarer medium is 90 degree. If a light ray is incident on the surface of a rarer medium 2 from a denser medium 1, then \[Sin(i_c) = n_{21}\]Here $ n_{12}$ is the refractive index of 1^st (denser) medium with respect to the 2^nd (rarer) medium.

24. Total internal reflection is the phenomenon of complete reflection of light back into the denser medium, when a light ray coming from denser medium is incident on the surface of a rarer medium.Two essential conditions for total internal reflection are:

(i)  The light ray should travel in a denser medium towards a rarer medium.

(ii)  Angle of incidence in the denser medium should be greater than the critical angle for the pair of media in contact.

25. Values of critical angle of glass-air and water-air interfaces are 41.5 degree and 48.75 degree, respectively.

26. The brilliance of diamond, action of optical fibres and mirage etc., are the phenomena based on total internal reflection of light.

27. Prism make use for total internal reflection phenomenon to bend light by 90 degree or by 180 degree or to invert images without changing their size. Such prism have one angle 90 degree and the other two angles 45 degree each and are known as totally reflecting prisms or poroprisms.

28. For refraction at a single spherical surface, all distances are measured from the pole of the refracting surface. The distances measured in the direction fo incidence of light are taken as positive and the distances measured in the opposite direction are taken as negative. If object is considered to be situated on left side of pole,  then the sign convention agrees with the cartesian coordinate system. Accordingly, all distances on left side of pole are taken as negative and on right side of pole as positive. The height measured above the principle axis are taken as positive as heights measured downwards are taken as negative.

29. For refraction at a single spherical surface \[\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2 - n_1}{R}\]Where light beam is going from medium of refractive index n1 to medium of refractive index n2. The relation is true for concave as well as convex spherical surfaces and irrespective of the fact whether refraction is taking places from rarer medium to denser medium or vice-versa.

30. According of lens maker’s formula \[\frac{1}{f}=(n_{21}-1)(\frac{1}{R_1}-\frac{1}{R_2})\]Where $n_{21}$ is the refractive index of lens material w.r.t. the surrounding medium, $R_1$ and $R_2$ are the radii of curvature of two surfaces of lens and f its focal length.

31. For image formed by a thin lens, we have \[\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\]All the above relation are the true for convex as well as concave surfaces/lenses and for real as well as virtual images.

32. Linear magnification (m) produced by a lens is defined as the ratio of the linear (lateral) size of the image to that of the object. Thus, \[m = \frac{h'}{h}=\frac{v}{u}=\frac{f-v}{f}=\frac{f}{f+u}\]For erect and virtual image, m is positive but for an inverted and real image, m is negative.

33. Power of lens is a measure of a degree of convergence or divergence of light incident on it. Mathematically, the power(p) of a lens is defined as the tangent of the angle by which it converges/ diverges a beam of light falling at unit distance from the optical centre. For a thin lens power is found to be the reciprocal of its focal length (f) i.e., \[P = \frac{1}{f}\]SI unit of power is dioptre (D).

34. Power of a converging (convex) lens is taken to be positive but that of a diverging (concave) less is taken negative.

35. For a combination of two (or more) thin lenses in contact, the effective focal length of the combination is given by \[\frac{1}{f} = \frac{1}{f_1}+\frac{1}{f_2}+......\]And in term of power, we have $ P = P_1 + P_2 + ...... $. 

36. For a combination of two or more lenses, the effective magnification for the combination is given by \[m = m_1 \times m_2 \times m_3 .......\]

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Electromagnetic Waves

1. A time-varying magnetic field gives rise to an electric field. Maxwell argued that a time-varying electric field should also give rise to a magnetic field. Maxwell thus tried to apply Ampere’s circuital law to find magnetic field outside a capacitor connected to a time-varying current. However, he noticed an inconsistency in Ampere’s circuital law.

2. To remove the inconsistency of Ampere’s circuital law, Maxwell suggested the existence of  "Displacement current”.

3. Displacement current ($ I_d$) is currents which come into play whenever the electric field and, consequently, the electric flux is changing with time. Mathematically, \[I_d = \epsilon_0 \frac{d\phi_E}{dt}\]

4. The sum of conduction current (I) and displacement current ($ I_d $) has the property of continuity along any closed path, although individually they may not be continuous. Thus, Maxwell modified Ampere’s circuital law as \[\oint \vec{B}.\vec{dl} = \mu_0 (I+I_d)\]With this modification the problem of inconsistency observed by Maxwell was rectified.

5. Maxwell was the first person who theoretically predicted the existence of electromagnetic waves, which are coupled with time-varying electric and magnetic fields propagating in space. The speed of these waves in free space is the same as that of light i.e. $ 3 \times 10^8 $ m/s.

6. Electromagnetic waves are produced by accelerated charges (or oscillating charge). An oscillating charge, which is an example of accelerating charge produces an oscillating electric field in space, which produces an oscillating magnetic field, which in turn is a source of oscillating electric field and so on. The oscillating electric filed and magnetic fields, thus, regenerate each other i.e., electromagnetic wave propagates through the space.

7. The frequency of the electromagnetic wave is same as the frequency of oscillation of the charge (electric field E) or the frequency of oscillating magnetic field (B).

8. Hertz was the first scientist to experimentally demonstrate the production of electromagnetic waves employing a crude form of an oscillatory LC circuit arrangement. Later on, Jagdish Chandra Bose produced electromagnetic waves of much shorter wavelengths. Marconi succeeded in transmitting electromagnetic waves over a distance of many kilometers.

9. Electromagnetic waves do not require any material medium for their propagation. In free space, their speed is given by \[c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8\]In a medium of absolute permittivity (), the speed of electromagnetic waves is given by \[c = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{K\mu_r}}\]

10. In an electromagnetic wave and electric and magnetic fields are in phase with each other. They attain their peak values at the same instant.

11. Electromagnetic waves are transverse in nature. The oscillating electric and magnetic fields are perpendicular to each other as well as perpendicular to the direction of propagation of the wave. In fact, the direction of ($ \vec{E}\times \vec{B} $) gives the direction of propagation of e.m. waves.

12. If we consider an electromagnetic wave propagating along positive x-axis then oscillating electric and magnetic fields may be represented as:\[\vec{E_y} = E_0sin(kx-\omega t)\hat{j}\] and \[\vec{B_z} = B_0sin(kx-\omega t)\hat{k}\]Here $\omega = 2\pi \nu $ is the angular frequency and $k = (\frac{2\pi}{\lambda})$ propagation constant of given electromagnetic wave.

13. In an electromagnetic wave, Amplitudes $E_0$ and $B_0$ of electric and magnetic fields in free space are related as: \[\frac{E_0}{B_0} = c\]

14. The energy density i.e., energy per unit volume of an electromagnetic wave consists of electric and magnetic contributions. Thus, The mean energy density \[U_m = U_E + U_B = \frac{1}{2}\epsilon_0 E^2_{rms} + \frac{1}{2\mu_0}B^2_{rms}\] It is found that average values of $ U_E $ and $ U_B $ are equal. 

15. Intensity of the electromagnetic wave is defined as the mean amount of energy passing through a unit area normally in unit time. It can be shown that Intensity \[I = U_m c = \frac{1}{2}\epsilon_0 c E^2_0 = \frac{c}{2\mu_0}B^2_0\]

16. The electromagnetic wave carries momentum too. If U be the total energy transferred to a surface by an electromagnetic wave in time t, then momentum delivered to this surface, assuming the surface to be completely absorbent, is \[p = \frac{U}{c}\]The average force exerted by e.m. wave on the surface will be \[F= \frac{p}{t} = \frac{U}{ct}\]

17. The classification of electromagnetic radiation waves according to frequency is known as “electromagnetic spectrum”. There is no sharp division between one kind of wave and the next and the classification is based roughly on how the waves are produced/ detected.

18. Complete electromagnetic spectrum in ascending order of frequency (or in decreasing order of wavelength) broadly consists of seven parts namely 

(i) Radio waves, (ii) Microwaves (iii) Infrared waves, (iv) Visible light rays, (v) Ultraviolet rays, (vi) X-rays, and (vii) Gamma rays.

19. Radio waves are produced by accelerated motion of charges in conducting wires and are used in radio and TV communication. They are in the frequency range of 500kHz to about 1000 MHz (or 1 GHz). These are further subdivided as a medium band, short band, HF band, VHF band, UHF band, etc.

20. Microwaves are extremely short-wavelength radio waves having a frequency range of $ 10^9 $ Hz to about 10 11 Hz and are produced by special vacuum tubes e.g., klystrons, magnetrons, and Gunn diodes. These are used in radar, microwave telecommunication, microwave oven, etc.

21. Inferred waves are produced by hot bodies and molecules and are characterized by their heating property. Inferred radiation plays an important role in maintaining the earth’s warmth by the greenhouse effect. Inferred rays are widely used in the remote switches of household electronic systems such as TV sets, video recorders, hi-fi systems, etc.

22. Visible parts are that part of the electromagnetic spectrum which is detected by the human eye. It runs from about $ 4 \times 10^{14} $ Hz to $ 7 \times 10^{14} $ Hz. Visible light emitted or reflected from objects around us provides us information about the world.

23. Ultraviolet rays consist of radiation in the frequency range $ 7 \times 10^{14} $Hz to $ 5 \times  10^{17} $ Hz (or wavelength range from 400 nm to 0.6 nm). These are produced by the sun, special lamps like mercury lamp, hydrogen tube etc, and very hot bodies. Ultraviolet rays have various uses such as in  LASIK eye surgery, to kill germs in water purifiers, as a disinfectant in hospitals, etc. however, ultraviolet light in large quantities has harmful effects on humans.

24. Ozone layer present in the atmosphere at an altitude of about 40 – 50 km absorbs most of the ultraviolet rays coming from the sun and thus, form a protective ring around the earth.

25. X-rays cover wavelengths from about 1 nm to $10^{-3} $ nm. These are produced by bombarding high energy electrons on a metal target. X-rays are used as a diagnostic tool in medicine, as a treatment for certain forms of cancer, and for scientific research.

26. Gamma rays are the hardest electromagnetic waves having wavelengths even less than $ 10^{-3} $ nm. These are produced in nuclear reactions and are also emitted during radioactive decay of the nuclei. These are used in medicine for destroying cancer cells.

Electromagnetic Spectrum

Type	Wavelength range	Production	Detection
Radio	> 0.1 m	Rapid acceleration and decelerations of electrons in aerials	Reciever's aerial
Microwave	0.1 m to 1 mm	Klystron valve or magnetron valve	Point contact diode
Infrared	1 mm to 700 nm	Vibration of atoms and molecules	Thermopiles, Bolometer, Infrared photographic film
Light	700 nm to 400 nm	Electrons in atom emit light when they move from one energy level to a lower energy level	The eye Photocells, Photographic film
Ultraviolet	400 nm to 1 nm	Inner shell electrons in atoms moving from one energy level to lower level	Photocells, Photographic film
X-ray	1 nm to $10^{-3}$ nm	X-ray tubes or inner shell electrons	Photographic film, Geiger tubes, Ionisation chamber
Gamma Ray	< $ 10^{-3} $ nm	Radioactive decay of the nucleus	- do -

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Newton's Laws and Force

1. Newton’s three laws of motion form the basis of mechanics. According to Ist law, A body continues to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state. This law defines force and is also called law of inertia.

According to second law, the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place in the direction of the applied force. This law gives us a measure of force. i.e. $ F \propto \frac{d\vec{p}}{dt} $. 

According to third law, To every action, there is always an equal and opposite reaction. This law gives us the nature of force.

2. Inertia is the inability of a body to change by itself, its state of rest, or its state of uniform motion along the straight line. Inertia is of three types: (i) Inertia of rest (ii) Inertia of motion, (iii)Inertia of direction.

3. From Newton’s 2^nd law, we obtain $ \vec{F_{ext}} = m \vec{a} $ i.e. an external force is the product of mass and acceleration of the body.

4. The absolute unit of force on SI in newton (N) and on cgs system, it is dyne. 

5. According to the principle of conservation of linear momentum, the vector sum of linear momentum of all the bodies in an isolated system is conserved and is not affected due to their mutual action and reaction. An isolated system is that on which no external force is acting. In other words, If external forces acting on the system is zero then it's linear momentum is constant.  Flight of rockets, jet planes, recoiling of a gun, etc. are explained on the basis of this principle. Newton’s 3^rd law of motion can also be derived from this principle and vice-versa.

6. Apparent weight of a man in an elevator is given by $ W' = m(g \pm a) $ where mg is real weight of the man. Acceleration is (+ a), when the lift is accelerating upward and (-a) when the lift is accelerating downwards. When lift is moving uniformly (upwards/downwards). a = 0. W’ = m g = real weight. In free fall, a = g,  W' = m (g – g) = 0 i.e. apparent weight becomes zero.

7. When two bodies of masses m₁ and m₂ are tied at the ends of an inextensible string passing over a light frictionless pulley, acceleration of the system is given by, \[a = \frac{|m_1 - m_2|}{m_1+m_2}g\], Tension is given by, \[T = \frac{2m_1m_2}{m_1+m_2}g\]

8. Impulse \[\vec{I} = \vec{F_{av}} \times t = \vec{P_2}-\vec{P_1}\] where t is the time for which average force acts $ (\vec{P_2 } – \vec{P_1})$ is change in linear momentum of the body.

9. The force which are acting at a point are called concurrent forces. They are said to be in equilibrium when their resultant is zero.

FRICTION

10. Friction is the opposing force that comes into play when one body is actually moving over the surface of another body or one body is trying to move over the surface of the other. Two causes of friction are: the roughness of surfaces in contact; Force of adhesion between the molecules of the surfaces in contact.

11. Limiting friction is the maximum value of static friction. Dynamic/Kinetic friction is somewhat less than the force of limiting friction.

12. Static friction is a self adjusting force.

13. Rolling friction is less than sliding friction.

14. Laws of limiting friction are: 

(i) $ F \propto R$, where R is normal reaction and F is the friction force.

(ii) Direction of F is opposite to the direction of motion.

(iii) F does not depend upon the actual area of contact.

(iv) F depends upon the nature of material and nature of polish of the surfaces in contact.

15. Coefficient of friction is given by, $ \mu  = \frac{F}{R} $.

16. Angle of Repose ($ \theta $) is the minimum angle of inclination of a plane with the horizontal, such that a body placed on the plane just begins to slide down.

17. Acceleration of the body down a rough inclined plane, \[a = g(sin\theta - \mu cos\theta)\]

18. Work done in moving a body over a rough horizontal surface, \[W = \mu mgd \]Work done in moving a body over a rough inclined plane, \[W = mg(sin\theta + \mu cos\theta)d\]

19. Friction is a necessary evil. Some of the methods of reducing friction are polishing, lubrication; streamlining the shape etc.

20. Centripetal force is the force required to move a body uniformly in a circle. The magnitude of this force is $ F = \frac{mv^2}{r}=mr\omega^2 $. It acts along the radius and towards the centre of the circle.

21. Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path. Centrifugal force can be treated as the reaction of centripetal force. The magnitude of centrifugal force is same as that of centripetal force. The direction of centripetal force is along the radius and away from the centre of the circle.

22. While rounding a level curved road, the necessary centripetal force is provided by the force f friction between the tyres and the road. The maximum velocity with which a vehicle can go round a level curve without skidding is $ v = \sqrt{\mu rg}$. To avoid dependence on friction, curved roads are usually banked i.e. outer edge of the curved road is raised suitably above the inner edge. If θ is the angle of banking, then $ tan\theta = \frac{v^2}{rg}$.

23. While rounding a banked curved road, the maximum permissible speed is given by \[v_{max} = \sqrt{\frac{rg(\mu_s + tan\theta)}{(1-\mu_s tan\theta)}}\]When frictional force is ignored, the optimum speed is, \[v_{max} = \sqrt{rg tan\theta }\].

24. Motion along a vertical circle is a non-uniform circular motion. Tension in the string at any position is $ T = \frac{mv^2}{r} + mgcos\theta $ where θ is the angle with vertical line through the lowest point of the circle.

1.                   For looping the vertical loop, the velocity of projection at lowest point L is $ v_L \geq \sqrt{5rg}$.

2.                   The value of velocity at the highest point H is $ v_H \geq \sqrt{rg}$.

3.                   Difference in tension in the string at lowest point and highest point of vertical circle is, $ T_L - T_H = 6mg $.

4.                   For oscillation over the arc of vertical circle $ 0 < v_L \leq \sqrt{2rg} $.

5.                   For leaving the vertical circle somewhat between $ 90^{\circ} < \theta < 180^{\circ} $, $ \sqrt{2rg} < v_L < \sqrt{5rg} $.

6.                   The minimum height h through which a motor cyclist has to descend to loop a vertical loop of radius r is, $ h = \frac{5}{2}r $.

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Alternating Current

1. An alternating current (a.c.) is that current which changes continuously in its magnitude and periodically reverses its direction. In its simplest form an a.c. may be sinusoidal function of time and may be expressed as  \[I = I_0sin(\omega t) = I_0sin(2\pi f t)\]where f is the frequency. 

2. Similarly an alternating voltage may be expressed as \[V = V_0sin(\omega t) = V_0sin(2\pi f t)\]

3. Mean (or voltage) value of an alternating current (or voltage) is zero for a whole (complete) cycle.

4. The root mean square (rms) or effective value of an a.c. is that steady current which, when passed through a resistance, produced exactly the same amount of heat is given time as is produced by actual a.c. when flowing through the same amount of heat in given time as is produced by actual a.c. when flowing through the same resistance for same time. It can be show that \[I_{rms} = \frac{I_0}{\sqrt{2}} = 0.707I_0\]\[V_{rms} = \frac{V_0}{\sqrt{2}} = 0.707V_0\]It is also sometimes referred as virtual value.

5. In an a.c. circuit, unless otherwise specified, we talk in terms of arms values of current and voltage.

6. When an alternating voltage $ V = V_m sin(\omega t) $ is applied to a pure resistor, the current flowing through the resistor is \[I = \frac{V_0}{R}sin(\omega t) = I_0sin(\omega t)\]where $ I_0 = \frac{V_0}{R} $. This means current is in the phase with the applied voltage.

7. When an alternating voltage $ V = V_0 sin(\omega t)$  is applied to a pure inductor (whose resistance is zero is 0) of inductance L, the current flowing through the inductor is \[I = \frac{V_0}{X_L}sin(\omega t - \frac{\pi}{2}) = I_0sin(\omega t - \frac{\pi}{2})\]where $ X_L = \omega L = 2\pi f L $ is called the inductive reactance of the given circuit. Unit of inductive reactance $X_L$ is a ohm $\Omega $. Moreover, current in a pure inductor lags behind the voltage by a phase angle $\frac{\pi}{2}$.

8. When an alternating voltage $ V = V_0 sin(\omega t) $ is applied to a pure capacitance C, the current flowing through the capacitor is \[I = \frac{V_0}{X_C}sin(\omega t + \frac{\pi}{2}) = I_0sin(\omega t + \frac{\pi}{2})\]Where $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ is called the capacitive reactance of given capacitor. Unit of capacitive reactance $X_C$ is ohm . Moreover, current in a capacitive circuit is ahead in phase than the voltage by a phase angle $\frac{\pi}{2}$.

9. To facilitate the analysis of an a.c. circuit we use a phasor diagram. A phasor is a vector which rotates about the origin with an angular speed. Amplitudes of phasors V and I represent the peak value $V_0$ and $I_0$ and the vertical components of phasors give the instantaneous values of voltage and current.

10. A pure inductor offers no opposition for flow of d.c. but offers an inductive reactance for flow of a.c. Magnitude of inductive reactance is directly proportional to the frequency of a.c.

11. A pure capacitance does not allow d.c. to pass through it but allows a.c. to pass through it. Value of capacitive reactance is inversely proportional to the frequency of a.c.

12. In an alternating current, circuit containing LCR in series, the potential difference may be added by the rule of phasors. As for a given current I, the voltage $V_L$ is ahead in phase by $\frac{\pi}{2}$, $V_R$ is in phase and $V_C$ lags behind in phase by $\frac{\pi}{2}$, hence resultant voltage will be given by \[V = \sqrt{V_R^2 + (V_L-V_C)^2}\]

13. Total opposition offered by an a.c. circuit for flows a current through it is called impedance and is denoted by Z. Its unit is ohm. Impedance plays the same role in an a.c. circuit, which is being played by resistance in a d.c. circuit. Thus, in an a.c. circuit, \[Z = \frac{V}{I} = \frac{V_{rms}}{I_{rms}} = \frac{V_0}{I_0}\]

14. When an alternating voltage $V = V_0 sin (\omega t)$ is applied to a LR series circuit, the current in the circuit is given by \[I = \frac{V_0}{\sqrt{R^2 + X_L^2}}sin(\omega t - \phi)=\frac{V_0}{Z}sin(\omega t - \phi) = I_0sin(\omega t - \phi)\]Where $ Z = \sqrt{R^2 + X_L^2} $ is the impedance of the circuit and current lags behind the voltage by a phase angle $ \phi $, given by $tan\phi =\frac{X_L}{R} = \frac{\omega L}{R}$.

15. When an alternating voltage $V = V_0 sin (\omega t)$ is applied to a RC series circuit, the current in the circuit is given by \[I = \frac{V_0}{\sqrt{R^2 + X_C^2}}sin(\omega t + \phi)=\frac{V_0}{Z}sin(\omega t + \phi) = I_0sin(\omega t + \phi)\]where $Z = \sqrt{R^2 + X_C^2} $ is the impedance of the circuit and voltage lags behind the current by a phase angle $ \phi $, given by $tan\phi =\frac{X_C}{R} = \frac{1/(\omega C)}{R}$.

16. When an alternating voltage $V = V_0  sin(\omega t)$ is applied to a LCR series circuit, the current in the circuit is given by \[I = \frac{V_0}{\sqrt{R^2 + (X_L-X_C)^2}}sin(\omega t - \phi)=\frac{V_0}{Z}sin(\omega t - \phi) = I_0sin(\omega t - \phi)\]where $Z = \sqrt{R^2 + (X_L-X_C)^2} $ is the impedance of the circuit and current lags behind the voltage by a phase angle $ \phi $, given by $tan\phi =\frac{X_L-X_C}{R} = \frac{\omega L - 1/(\omega C)}{R}$.

17. The average power dissipated in an a.c. circuit is given by \[P_{av}=I_{rms}V_{rms}cos(\phi)\]Where $\phi$ is the phase angle between voltage and current. The term ‘$cos(\phi)$’ is referred  as the power factor. Here following special cases arise :

(i)    For a pure resistive circuit $ P_{ av} = V_{ rms} I_{ rms} $

(ii)   For a pure inductive or a pure capacitance circuit, power factor $cos (\phi)$ has a zero value and hence net power consumed over an entire cycle of a.c. is zero. Such type of electrical circuit is known as a “wattles circuit” and current flowing is known as “wattles current”.

18. In a LCR series circuit, in general, \[I_0 = \frac{V_0}{\sqrt{R^2 + (X_L-X_C)^2}}=\frac{V_0}{Z}\]If as a special case $X_L = X_C$, then Z = R = a minimum and consequently the current amplitude $ I_0 = \frac{V_0}{R} $= a maximum and current and voltage are in same phase. Such a situation is called “electrical resonance”. Resonance takes place when $X_L = X_C$ or when angular frequency \[\omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi \sqrt{LC}}\]

19. The quality factor (Q factor) of a resonant circuit is a measure of the “sharpness of resonance” and is defined as the ratio of resonant angular frequency $\omega_0$ to the band width $(2\Delta \omega )$ of the circuit, where band width is the difference in angular frequencies $(\omega_0 - \Delta \omega)$ and $(\omega_0 + \Delta \omega ) $ at which power is half the maximum power or current is $(\frac{1}{\sqrt{2}})$ times the maximum current value at resonance. Mathematically, \[Q = \frac{\omega_0}{2\Delta \omega} = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 CR} = \frac{1}{R}\sqrt{\frac{L}{C}}\]The quality factor is large if resistance R is low or inductance L is high. High quality factor or high sharpness of resonance means high selectivity and the tuning of the circuit for resonance will be better.

20. When a capacitor (initially charged) is connected to an inductor, the charge on the capacitor and the current in the circuit, exhibit electrical oscillation just like a harmonic oscillator. The angular frequency and the frequency of these oscillations are \[\omega_0 = \frac{1}{\sqrt{LC}}\]

21. For an ideal L – C circuit there is no dissipation of energy and amplitude of oscillations remains constant. Energy in the system oscillates between the capacitor and the inductor. Average value of electrostatic energy and of magnetic energy is same and total electromagnetic energy \[u = \frac{1}{2}\frac{q^2}{C} + \frac{1}{2}LI^2\]However, practically oscillations are damped one due to two reasons, namely (i) presence of some resistance in the inductor, and (ii) radiation in energy in the form of electromagnetic waves.

22. A transformer is a device used in a.c. circuits to change the voltages. A transformer which increases the a.c. voltage is called ‘set-up’ transformer, where as the ‘step-down’ transformer reduce the a.c. voltage.

23. A transformer works on the principle of mutual induction and consists of a primary coils and a secondary coil wound on a laminated soft iron core. It is found that for an ideal transformer (in which there is no loss of electrical energy), we have \[\frac{V_s}{V_p} = \frac{I_p}{I_s} = \frac{N_s}{N_p}= k\]Where $N_s$ and $N_p$ are the number of turns in windings of secondary and primary coils and k is called the transformation ratio. In step up transformer, $ V_s > V_p, N_s > N_p, I_s < I_p$. while in step down transformer, $ V_s < V_p, N_s < N_p, I_s > I_p$.

24. In a set up transformer there is some loss energy and hence output given by transformer is less than the input supplied to it. Four main causes of energy losses in a transformer are (i) magnetic flux leakage, (ii) resistance of the windings, (iii) eddy currents, and (iv) magnetic hysteresis. However, by taking appropriate preventive measures, these energy losses can be minimised and controlled.

25. Generally, a.c. power is transmitted from one station to another at highest possible voltage so that line current is less and consequently power loss during transmission is least possible. It is achieved by use of step up transformers at the generating station. At the consumer station, using step down transformers, the power is again supplied to homes and establishments at comparatively low voltages.

Watch Video Lectures

4.6 A.C. Generator Fully explained watch video

4.7  Mean and Root mean square values of Alternating Current and voltage watch video

4.8 A.C. through Resistance and Inductor coil watch video

4.9 A.C. through Capacitors and RLC in series watch video

4.10 A.C. through RL, RC | LC Oscillations and Resonance condition watch video

4.11 Quality Factor and Power dissipation in Resistance, Inductor and Capacitor watch video

4.12 Power dissipation in RLC circuit and Power Factor watch video

4.13 Transformer and Choke Coil watch video

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Electromagnetic Induction

1. Electromagnetic induction is the phenomenon of production of electric emf (or current) in a circuit whenever the magnetic flux linked with the circuit changes. In other words, electromagnetic induction is the phenomenon in which electric current is generated in a circuit by varying magnetic fields in that region.

2. The magnetic flux linked with a surface held in a magnetic field is defined as the total number of magnetic field lines crossing the surface normally. Mathematically, magnetic flux linked with a surface \[\Phi_B = \vec{B}.\vec{s} = \int \vec{B}.\hat{n}ds = \int Bcos\theta ds\]

3. Magnetic flux is a scalar quantity and its SI unit is 1 weber (1 Wb). One weber is the magnetic flux linked with a surface area of $ 1 m^2 $ when held normally inside a uniform magnetic field of 1 tesla. (Thus, $ 1 Wb = 1 T m^2 $).

4. On the basis of his experimental studies, Faraday gave following laws of electromagnetic induction: Whenever there is a change of magnetic flux through a circuit, there will be an induced emf and this emf will last as long as the change persists. The magnitude of the induced emf is equal to the time rate of change of magnetic flux through the circuit. Mathematically, the induced emf e is given by \[\varepsilon = - \frac{d\Phi_B}{dt} = \frac{\phi_i - \phi_f}{t}\]Here -ve sign indicates the direction of induce emf and hence the direction of induced current in a closed loop.

5. For a coil of N-turns of same cross-sectional area, we have \[\varepsilon = - N\frac{d\phi_B}{dt} = - \frac{d}{dt}(N\phi_B)\]

6. If an electrical circuit is complete, an induced current flow in the circuit on account of the induced emf. Magnitude of induced current is given by \[I = \frac{\varepsilon}{R} = - N\frac{d\phi_B}{Rdt}\] 

7. As magnetic flux $\phi_B $ can be varied by changing either the magnetic field B or by changing the area of the coil/circuit or by changing the shape of a coil or rotating a coil in a magnetic field such that angle $\theta $ between B and s changes, hence induced emf can be set up by any of these methods.

8. Lenz’s law gives the direction of induced emf/current. According to it, the polarity of the induced emf is such that it tends to produced a current which opposes the change in magnetic flux that produced it.

9. The phenomenon of electromagnetic induction and Lenz’s law are strictly in accordance with the principle of conservation of energy.

10. Lenz’s law is usually applied to know the direction of induced currents for closed circuits 

11. Motional emf is the emf induced in a conductor, when it is moving in the direction perpendicular to its length and a uniform time-independent magnetic field is present which is perpendicular to the plane of the conductor as well as the direction of motion. If a conducting rod of length ‘l’ moves with a constant speed ‘v’ in a normal uniform magnetic field ‘B’, the magnitude of motional emf is given by Motional emf \[|\varepsilon | = Blv\]

12. Whenever magnetic flux linked with a bulk metallic conductor changes, induced currents are set up in the conductor in the form of closed loops and are, thus, known as eddy currents. Hence, eddy currents are the currents induced in a bulk conductor when placed in a changing magnetic field.

13. Eddy currents are undersirable since they dissipate electric energy in the form of heat. To reduce eddy currents (i) slots are made in the conductor, and (ii) the conducting parts are built in the form of laminated metal sheets separated by an insulating lacquer. The plane of the laminations are arranged parallel to the magnetic field.

14. Eddy currents cause electromagnetic damping which may be used in (i) electric brake system, (ii) induction furnace, (iii) speedometer, (iv) electromagnetic damping, (v) moving coil galvanometer to make it dead beat, (iv) a.c. induction motor etc.

15. Self -induction is the phenomenon according to which an opposing induced emf is produced in a coil, as a result of change in current flowing through it. Self-induction is also referred to as the “electrical inertia”.

16. The coefficient of self-induction or self-inductance (L) of a coil is numerically equal to the magnetic flux linked with the coil, when a unit current flows through it.

17. The self-inductance of a coil depends only on the geometry of the coil and intrinsic material properties. Moreover, inductance is a scalar quantity.

18. Value of induced emf due to self-induction phenomenon is given by\[\varepsilon = -L\frac{dI}{dt}\]

19. Self-inductance of a coil is numerically equal to the induced emf produced in the coil, when rate of change of current in the coil is unity.

20. SI unit of self-inductance is 1 henry (1 H). Inductance of a coil is said to be 1 henry, if a rate of change of current of $A s^{-1}$ induces an emf of 1 volt in it.

21. The self-inductance of an air core long solenoid of length l, total number of turns N and cross-section area A is given by \[L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 lA\]where n is the number of turns per unit length. 

22. Mutual induction is the phenomenon according to which an opposing emf is induced in a coil, as a result of change in current or magnetic flux linked with a neighboring coil.

23. Coefficient of mutual induction or mutual inductance (M) of a pair of two neighboring coils is numerically equal to the magnetic flux linked with one coil when a unit current flows through the neighbouring coil.

24. Induced emf due to mutual induction phenomenon is given by \[\varepsilon _1 = - M\frac{dI_2}{dt}\]Hence, mutual inductance for a given pair of two coils is numerically equal to the induced enf produced in one coil when the rate of change of current in the other coil is unity.

25. SI unit of mutual inductance too is 1 henry (1H).

26. Mutual induction of the pair of coaxial long solenoids is given by \[M = \mu_0 \mu_r \frac{N_1N_2A}{l} = \mu_0 \mu_r n_1n_2lA\]where $ n_1 , n_2 $ is the number of the turns per unit length. The mutual inductance of a pair of coils also depends on their separation as well as their relative orientation.

27. If there are two solenoids $S_1$ and $ S_2 $ then it can be easily proved that  mutual inductance $ M_{21} $ of solenoid $ S_2 $ with respect to $ S_1 $ is exactly equal to mutual inductance $ M_{12} $ of solenoid $ S_1 $ with respect to $ S_2 $ i.e.,  \[M_{12} = M_{21} = M\]

28. The self-induced emf is also called back emf as it opposes any change in the current in a circuit. Thus, work is to be done against the back emf in establishing a current in the coil. The work done in establishing a current I in a coil of inductance L is given by \[W = \frac{1}{2} LI^2\]

29. Energy stored (in the form of magnetic energy) in an inductor L, while a current I is established in it, is given by \[U = \frac{1}{2} LI^2\]

30. The magnetic energy stored per unit volume (or the magnetic energy density) in a region of uniform magnetic field ‘B’ is usually given  \[u = \frac{B^2}{2\mu_0}\]

31. When a conducting rod of length l kept perpendicular to a uniform magnetic field B is rotating about one of its ends with a uniform angular velocity $ \omega $, the emf induced between its ends has a magnitude \[\varepsilon = \frac{1}{2}B\omega l^2\]However, when the rod is rotating about its centre, there is no emf induced between its ends.

32. If a flat rectangular coil of N-turns each of area A is rotating in a uniform magnetic field B with a uniform angular velocity $\omega $ so that its axis of rotation is in the plane of the loop and is at right angles to the magnetic field, the induced emf at any instant t is given by the relation, Induced emf \[\varepsilon = N B A\omega sin(\omega t)\]

33. Thus, whenever the coil is perpendicular to the magnetic field, magnitude of induced emf is zero and whenever the coil is parallel to the magnetic field, magnitude of induced emf is zero and whenever the coil is parallel to the magnetic field, the magnitude of induced emf is maximum having a value\[\varepsilon_{max} = N B A\omega \]

34. The direction of induced emf is (current) in a coil when rotated in a uniform magnetic field may be easily obtained by Fleming’s right-hand rule. According to it, stretch the central finger, the fore-finger point and the thumb of your right hand mutually perpendicular to each other such as the for-finger points in the direction of magnetic field and thumb toward the motion of conductor then the central figure points in the direction of induced current (emf) in the conductor. Fleming’s right-hand rule is in accordance with Lenz’s law.

35. An a.c. generator is a device that converts machinal energy into electric energy on the basis of electromagnetic induction.

36. An a.c. generator is based on the principle of production of induced emf in a rectangular coil, being rotated about its axis with a uniform angular velocity when a uniform magnetic field is present in a perpendicular plane. The induced emf \[\varepsilon = N B A\omega sin(\omega t)\] changes both in magnitude as well as direction with time and is, therefore, known as alternating emf (or induced current is known as an alternating current).

Watch Video Lectures

4.1 Magnetic Flux and Faraday's Law watch video

4.2 Lenz Law and Motional EMF watch video

4.3 Eddy current and self-induction watch video

4.4 Self Induction of solenoid and Grouping of the Inductor coils watch video

4.5 Mutual Induction and Mutual Induction of the solenoids watch video

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Matter and Magnetism

 1. Permanent magnets of different shapes may be prepared from iron, steel, nickel, cobalt, and their alloys.

2. A bar magnet or a magnetic dipole is the simplest type of a permanent magnet.

3. A bar magnet is characterized by:  Its the directional property,  Attracting magnetic materials towards it,  Including magnetism in other magnetic materials etc.

4. Magnetic monopoles are not found in nature.

5. The magnetic effects in magnetic material are due to atomic magnetic dipole in the materials. These dipoles result from the effective current loops of electrons in atomic orbits.

6. All magnetic phenomena can be explained in terms of circulating currents. A current loop of area A carrying a current I is equivalent to a magnetic dipole of dipole moment $ \vec{m} = I\vec{ A} $ along the axis of the loop. If there are N number of turns in a coil then $ \vec{m} = NI\vec{ A} $. SI unit of magnetic dipole moment is ampere-meter².

7. From the resemblance of magnetic field lines for a bar magnet and a solenoid, we may consider a bar magnet as a large number of circulating current like a solenoid. In fact, a bar magnet and a solenoid produce similar magnetic fields. The magnetic moment of a bar magnet is thus equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field.

8. Magnetic field lines are a visual and intuitive realization of the magnetic field. A magnetic field is a smooth curve in a magnetic field, tangent to which at any point gives the direction of the magnetic field at that point.

9. In free space around a magnetic dipole the magnetic field lines start from N-pole and end at S-pole. However, inside the magnet, they travel from S-pole to N-pole. Thus, magnetic field lines of a magnetic or a solenoid form continuously closed curves.

10. The magnetic field lines do not intersect one another. It is so since the direction of the magnetic field would not be unique at the point of intersection.

11. The larger the number of magnetic field lines crossing per unit normal area in a given region, the stronger is the magnetic field B there.

12. The magnetic field B at the point on the axial line of a bar magnet (magnetic dipole) of dipole moment m at a distance r from the mid-point of magnet is given by \[\vec{B} = \frac{\mu_0}{4\pi}\frac{2\vec{m}r}{(r^2-l^2)^2}\]for short dipole or where r >> l, \[\vec{B} = \frac{\mu_0}{4\pi}\frac{2\vec{m}}{r^3}\]Direction of $ \vec{B}$ is same as of $\vec{m}$. 

13. The magnetic field B at a point on the equatorial line of a magnetic dipole of magnetic moment m at a distance r from the mid-point dipole is \[\vec{B} = -\frac{\mu_0}{4\pi}\frac{\vec{m}}{(r^+l^2)^(3/2)}\]and for short dipole or when r >> l, \[\vec{B} = -\frac{\mu_0}{4\pi}\frac{\vec{m}}{r^3}\]Here -ve sign means that direction of magnetic field is opposite to the direction of dipole moment.

14.Torque acting on a magnetic dipole of moment M placed in a uniform magnetic field B is given by\[\vec{\tau} = \vec{M} \times \vec{B}\]and \[{\tau} = MBsin(\theta)\],where $\theta $ is is the angle between the magnetic axis of dipole and the magnetic field. The torque tends to align the magnetic dipole along the direction of magnetic field.

15. The potential energy of a magnetic dipole placed at angle $\theta $ with the magnetic field B is \[U = -\vec{m}.\vec{B} = -mBcos\theta\]where we choose the zero of energy at the orientation when m is perpendicular to B. For $ \theta $ = 0⁰, potential energy of a magnetic dipole is -mB and it corresponds to stable equilibrium state of magnetic dipole in a magnetic field. However, for $ \theta = \pi $, U = mB and it corresponds to unstable equilibrium of magnetic dipole.

16. A magnetic dipole freely suspend in a uniform magnetic field B, if once twisted by a small angle $\theta$ and then released, executes simple harmonic oscillations. The time period of oscillation is given by \[T = 2\pi \sqrt{\frac{I}{mB}}\]Where I = moment of inertia of magnetic dipole about the suspension axis and m = magnetic dipole moment.

17. According to Gauss’ law for magnetism, “ the net magnetic flux through any closed surface is zero” i.e., \[\Phi_B = \oint \vec{B}.\vec{dS} = 0\]It is so because in magnetism isolated monopoles do not exist. There are no source/ sink of magnetic field B. the simplest magnetic element is a dipole or a current loop.

18. Our earth has a magnetic field of its own. The earth’s magnetic field resembles that of a (hypothetical ) giant magnetic dipole which is aligned making a small angle with the rotational axis of the earth. Its magnetic north pole N_m is near the geographic south pole S_g and its magnetic south pole S_m is near the geographic north pole N_g. the earth’s magnetic field may be approximated by a dipole with magnetic moment 8.0 x 10²² A-m².

19. The strength of earth’s magnetic field varies from place to place on the earth’s surface. The magnitude of the field is of the order of 4 x10^-5 T.

20. Magnetic element of a place are three quantities needed to specify the magnetic field of the earth at the given place. The three magnetic element are (i) the magnetic declination, the magnetic dip, and (iii) the horizontal component of earth’s magnetic field.

21. Magnetic declination (D) at a place are the angle which magnetic meridian at that place subtends from the geographic meridian. Effectively, it is the angle between the true geographic north and the north shown by a compass needle.

22. Magnetic dip $\delta $ or angle of inclination is the angle in which direction of earth magnetic field at a place subtends from the horizontal direction along the magnetic meridian.

23. If $B_E$ be the magnetic field of earth at a given place and $\delta $ be the magnetic dip then horizontal component of earth magnetic field is $ B_H = B_Ecos\delta $ and the vertical component of earth field is $B_V = B_Esin\delta $. \[B_E^2 = B_H^2+B_V^2 \implies B_E = \sqrt{B_H^2+B_V^2}\] \[tan\delta = \frac{B_V}{B_H}\]

23. Earth magnetic field is thought to arise due to electrical produced by convective motion of metallic fluids (consisting mostly of molten iron and nickel) in the outer core of the earth. It is known as the ‘dynamo effect’.

24. Magnetic equator is the axis, at all points of earth’s magnetic field is directed horizontally i.e. $ B_E = B_V $ and angle of dip, as well as vertical component of earth’s magnetic field $B_V $, have zero value.  

25. At magnetic poles of earth $\delta = \frac{\pi}{2}$, $B_H= 0$ and $B_V=B_H $ value of dip angle gradually increases as one goes from equatorial region towards the poles of earth.

26. At the magnetic poles a compass needle may point along any direction. However a dip needle will point straight down at the magnetic poles.

27. In free space if magnetic field at a given place be $ \vec{B_0 } $ then we define a term known as” magnetic intensity” H as \[\vec{H} = \frac{\vec{B_0}}{\mu_0}\] where $ \mu_0 $ is the magnetic permeability of free space.

28. When a magnetic material is placed in a magnetic field $ B_0 $ the field changes to B on account of magnetization of that material. the net magnetic moment developed in the given material per unit volume is known as “magnetization” (or intensity of magnetization) M of that material. Thus \[\large \vec{M} = \frac{\vec{m_{net}}}{V}\]SI unit of magnetization $\vec{M}$ is A-m^-1.

29. In the presence of a magnetic material, the magnetic field change from $ \vec{B_0} $ to $ \vec{B}$ where \[\large \vec{B} = \mu_r \vec{B_0}\]and $ \mu_r $ is known as relative magnetic permeability of given material and is a unitless and dimensionless quantity.

30.  It is observed that \[\large \vec{B} = \vec{B_0} + \vec{B_m} = \mu_0\vec{H} + \mu_0\vec{M} = \mu_0(\vec{H} + \vec{M})\]

31. Magnetic susceptibility of a magnetic material $ \chi $ is defined as per relation\[\large \vec{M} = \chi \vec{H}\]It is a measure of how a magnetic material responds to an external magnetic field. Magnetic susceptibility is a unitless and dimensionless quantity. It is found that  \[\large \mu_r=1+ \chi\]

32. \[\large \mu_r . \mu_0 = \mu\]is the absolute magnetic permeability of given material. Units and dimensions of $ \mu$ are same as of $ \mu_r $

33. Diamagnetic materials are those which experience a feeble force of repulsion when placed in a strong external magnetic field. Diamagnetic substances tend to move from stronger to weaker part of the external magnetic field. The field lines are repelled or expelled and the field inside a diamagnetic material is reduced. The individual atoms of a  diamagnetic material do not possess a permanent magnetic dipole moment of their own but a small dipole moment in the opposite direction is developed in them when placed in an external magnetic field. Bismuth, copper, lead, nitrogen, water, etc., are diamagnetic in nature. For diamagnetic material $ -1 \leq  \chi $ <0, $ 0 \leq \mu_r < 1  $ and $\mu < \mu_0 $. A superconductor is a perfect diamagnetic for which $\chi = 0, \mu_r = 0, \mu_0 = 0 $.

34. Paramagnetic materials are those which experience a weak force of attraction when placed in an external magnetic field. Paramagnetic substances are weakly magnetized when placed in an external magnetic field. Field lines are attracted and the field inside a paramagnetic material is increased. They have a tendency to move from weaker to stronger regions of magnetic field.  The individual atoms possess a permanent dipole moment and this dipole moment tries to align itself in the direction of external field B₀. Aluminum, sodium, calcium, oxygen, etc., are paramagnetic. For paramagnetic materials, $ \chi $ is small positive, $\mu_r $ is greater than 1. 

35. Ferromagnetic materials are those which are strongly attracted by an external magnetic field and which can themselves be magnetized. Iron, nickel, cobalt and some of their alloys are ferromagnetic. For ferromagnetic materials $\chi >> 1, \mu_r >>1, \mu >> \mu_0    $.

36. The individual atoms in a ferromagnetic material possess a permanent dipole moment. These atomic dipoles interact with one another so as to form domains. Ordinarily, the magnetization varies randomly from domain to domain and net magnetization is zero. Under the influence of an external magnetic field the domains are aligned accordingly and the sample acquires magnetization.

37. Ferromagnetic materials are said to be hard if magnetization persists even after the removal of external magnetic field. Ferromagnetic materials are called soft it magnetization disappears on removal of external field.

38. According to Curie’s law magnetization $\vec{M} $ of a paramagnetic material is directly proportional to applied magnetic field B₀ and inversely proportional to the absolute temperature T. Thus, \[\large \vec{M} = \frac{C\vec{B_0}}{T}\]where C is known as the Curie’s constant. In terms of susceptibility, we have $ \chi = C \frac{\mu_0}{T} $.

39. The ferromagnetic property of a material gradually decrease as the temperature is raised. Above a certain “temperature is transition” (also known as Curie temperature) a ferromagnetic material begins to behave as a paramagnetic substance. The susceptibility above the Curie’s temperature is described by: \[\large \chi = \frac{C}{T- T_C}\]

40. Relation between B and H in ferromagnetic material is complex and represented by a hysteresis curve. The word hysteresis means lagging behind of B w.r.t. H. 

41. The residual magnetization of a ferromagnetic substance undergoing an hysteresis cycle must be subjected in order to demagnetize it completely, is known as ‘coercive force’ or ‘coercivity’. 

42. During a complete magnetization cycle of a material some energy is dissipated, which appears as heat. Area of B-H hysteresis loop gives the energy dissipation per unit volume per cycle. Steel has a wide hysteresis loop but soft iron has a narrow hysteresis curve. 

43. The hysteresis curve allows us to select suitable materials for a magnet. Material for a permanent magnet should have high retentivity ,high coercivity and a high permeability. Steel is a favoured  choice for permanent magnet. Material for an electromagnet should  have high permeability, low retentivity and a narrow hysteresis curve soft iron is therefore preferred for making an electromagnet.  

 

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