Work Energy Power Test1

 Work Energy Power Test

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2. Read the Questions carefully before choosing the option. 

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4. There is no negative marking in this test.

5. The time limit is only 20 minutes.

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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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Current Electricity (Previous Year Questions)

Q1.  heating element using nichrome connected to a 230 V supply draws an initial current of 3.2 A which settles after a few second to a steady value of 2.8 A. what is the steady temperature of the heating element if the room temperature is 27.0 ^oC ? temperature coefficient of nichrome average over the temperature range involved is 1.70 x 10^{-4 o}C^-1.

Ans. Here V = 230 V and at $T_1 = 27$ degree celcius, current   $ I_1 = 3.2 $A \[R_1 = \frac{V}{I_1} =\frac{230}{3.2} \Omega\]

Again at a steady temperature $ T_2$ of the heating element, current $I_2 = 2.8 $ A \[R_2 = \frac{V}{I_2} =\frac{230}{2.8} \Omega\]

Moreover temperature coefficient of resistance $ \alpha = (1.70 * {10^{-4}}) ^{\circ} C^{-1}$

Using the relation  \[R_2 = R_1 [1 + \alpha (T_2 -T_1)]\], we have \[T_2 -T_1 = \frac{R_2 - R_1}{R_1 \alpha} = 840\] \[T_2 = T_1 + 840 = 27 +840 = 867^{\circ}C\]

Q2. (a)    In a metre bridge, the balance point is found to be at 39.5 cm from the end A containing X toward end A, when the resistor Y is of 12.5$\Omega$. Determine the resistance of X. why are the connections between resistors in a wheatstone or meter bridge made of thick copper strips ?

(b)   Determine the balance point of the bridge above if X and Y are interchanged.

(c)    What happens if the galvanometer and cell are interchanged at the balance point of the bridge ? would the galvanometer shown any current?

Ans. (a) here Y = 12.5 ohm, length AD = $l_1 = 39.5$ cm \[\frac{X}{Y} = \frac{l_1}{100-l_1}\] \[X = Y\frac{l_1}{100-l_1} = 12.5*\frac{39.5}{60.5} = 8.2 \Omega\]

Connection are made of thick copper strips so that their resistance may be extremely small and negligible, because these resistances are not accounted for in the formula of meter bridge.

(b) let on interchanging X and Y, the new balance point is obtained at $l_2$, then\[\frac{Y}{X} = \frac{l_2}{100-l_2} \implies l_2 = 60.5 cm\]

(c) At the balance point at the bridge if the galvanometer at the cell are interchanged, it makes no effect on balance condition and the galvanometer will not show any deflection.

Q3. State the condition in which terminal voltage across a secondary cell is equal to its emf.

Ans. When the cell is in an open circuit i.e., when no current is being drawn from the cell.

Q4. Under what condition can we draw maximum current from a secondary cell?

Ans. When external resistance present in the circuit is zero i.e., when the cell is short circuited.

Q5. A wire of resistivity $\rho$ is stretched to twice its length. What will be its new resistivity?

Ans. Resistivity will remain unchanged, because resistivity of a material is independent of its dimensions.

Q6. A physical quantity, associated with electric conductivity, has the SI unit ‘’ohm-meter.”  Identify the physical quantity.

Ans. Resistivity.

Q7. Define electrical conductivity of a conductivity of the conductor and give its SI unit.

Ans. Reciprocal of resistivity of a conductor is called its conductivity. Alternatively conductance of a unit cube conductor is called its electric conductivity. Its SI unit is S m-1.

Q8. If potential difference V applied across a conductor is increased to 2 V , how will the drift velocity of the electrons change ?

Ans. Drift speed   \[v_d = \frac{eE}{m}\tau = \frac{eV}{ml}\tau\]Thus, it is clear that on increasing the potential difference from V to 2V, the drift speed of the electrons is doubled.

Q9. What is the effect of heating of a conductor of a drift velocity of a free electrons?

Ans. On heating a conductor its resistance increase or the current decreases. Consequently, the drift velocity of free electron decreases.

Q10. If the temperature of a good conductor increase, how does the relaxation time of electrons in the conductor change?

Ans. With increase in temperature the resistivity of conductor material increases and hence in accordance with the formula $ \rho = \frac{m}{ne^2\tau} $, the relaxation period $\tau$ decreases.

Q11. Two conducting wires X and Y of same diameter but different materials are joined in series across a battery. If the number density of electrons in X is twice that in Y, find the ratio of drift velocity of electrons in the two wires.

Ans. It is given that number density of electrons in X is twice that in Y, i.e., $ n_x = 2n_y $. As in a series circuit the electric current flowing through the entire circuit is exactly same, Hence \[I =n_xA_X e(v_d)_X = n_YA_Ye (v_d)_y\]As both wire have same diameter, hence $ A_x = A_y $ \[\frac{(v_d)_x}{(v_d)_y} =\frac{n_y}{n_x}= \frac{n_y}{2n_y} = 0.5\]

Q12. Two wires of equal length, one of copper and other of manganin have the same resistance. Which wire is thicker ?

Ans. In accordance with the formula $ R = \rho \frac{L}{A} $ for same resistance R and length l, \[A \propto \rho\]. Hence, the manganin wire will be thicker because its resistivity is more.

Q13. Write an expression for the resistivity of a metallic conductor showing its variation over a limited range of temperatures.

Ans. $\rho_T = \rho_0[ 1 + \alpha(T - T_0) ] $, where $\alpha$ is the temperature coefficient of resistivity.

Q14. Why are alloys, maganin and constantan used to make standard resistance coils ?

Ans. Because their resistivity is high and temperature coefficient of resistance is extremely small.

Q15. The metallic conductor is at a temperature $\theta_1$. The temperature of the metallic conductor is increased to $\theta_2$. How will the product of its resistivity and conductivity change ?

Ans. The product of resistivity and resistivity and conductivity always remains constant

Q16. The three coloured bands on a carbon resistor are red, green and yellow respectively. Write the value of its resistance.

Ans. Value of given resistance is $25*10^4 + 20%  \Omega $.

Q17. The sequence of bands marked on a carbon resistor are : Brown, black, green and gold. Write the value of resistance with tolerance.

Ans. Resistance R = $ 10^6 $ ohm $\pm$ 5%.

Q18. Which physical quantity does the voltage vs. current graph for a metallic conductor depict ? Give its SI unit

Ans. Electrical resistance is given by the slope of V – I graph. Its SI unit is a ohm.

Q19. A(i) series, (ii) parallel combination of two given resistor is connected, one by one, across a cell. In which case will the terminal potential difference, across the cell, have a higher value?

Ans. Terminal potential difference V = E – Ir, where r is the internal resistance of the cell. If two given resistor be $R_1$ and $R_2$ than in series $I_s = \frac{e}{(R_1 + R_2 + r)}$ but in parallel combination current $ I_p = \frac{e}{(\frac{ R_1R_2}{R_1+R_2})+ r }$. Obiviously, $I_s < I_p $. Hence Vs >Vp.

Q20. A cell of emf 2 V and internal resistance $0.1\Omega$ is connected to a $3.9\Omega $ external resistance. What will be the potential difference across the terminals of the cell?

Ans. Terminal potential difference \[V = \frac{eR}{R+r} =\frac{2*3.9}{3.9+0.1}= 1.95 V\]

Q21.What happens to the power dissipation if the value of electric current passing through  conductor of constant resistance in doubled?

Ans. In accordance with formula $ P = I^2R $, the dissipation becomes 4 times if the current passing through a given resistance is doubled.

Q22. Which has a greater resistance, 1 kW electric heater or a 100 W electric bulb, both marked for 200V? 

Ans. Electric bulb marked 220 V – 100W will have higher resistance because its power is less and power is given by $ P = \frac{V^2}{R} \implies R = \frac{V^2}{P} $.

Q23. Two bulb whose resistance are in the ratio of 1:2 are connected in parallel to a square of constant voltage. What will be the power dissipation in these?

Ans. Here V = constant and $\frac{R_1}{R_2} = \frac{1}{2} $, hence  $ \frac{P1}{P2} = \frac{V^2/R_1}{V^2/ R_2} = \frac{R_2}{R_1} = 2$.

Q24. A toaster produces more heat than a light bulb when connected in parallel to the 220 v mains. Which of the two has greater resistance ?

Ans. From the relation $ P = \frac{V^2}{R}$, it is clear that the resistance of bulb is greater as it produces less heat (i.e., its power is less) for constant potential difference.

Q25. Two bulbs are marked 60 W, 220 V, and 100 W, 220 V. These are connected in parallel to 220 V mains. Which one out of the two will glow brighter ?

Ans. Bulb marked 100W, 220V will glow brighter because its power is more.

Q26. Two conductors one having resistance R and another 2R are connected in turn across a d. c. source . If the rate of heat produced in the two conductors is $Q_1$ and $Q_2$ respectively, what is the value of $\frac{Q_1}{Q_2}$ ?

Ans. Here V = const., hence, $ \frac{Q_1}{Q_2} = \frac{R_2}{R_1} = \frac{2R}{R} = 2:1 $.

Q27. A heater joined in series with a 60 W bulb is connected to the mains. If 60 W bulb is replaced by a 100 W bulb, will the rate of heat produced by the heater be more, less or remain the same ?

Ans. We know that resistance of a 100W bulb is less than that of 60 W bulb. Hence, on joining 100 W bulb (instead of 60 W bulb) with heater, the resistance of the circuit decreases and consequently, circuit current increases. Hence, heat produced by the heater rises.

Q28. Two heater wires of the same dimensions are first connected in series and then in parallel to a source of supply. What will be the ratio of heat produced in the two cases ?

Ans. Let resistance of each heater be R then in series arrangement $R_S = 2R $ and in parallel arrangement $R_P = \frac{R}{2} $. In accordance with formula $ H = \frac{V^2t}{R} $, ratio of heat produced in two cases: \[\frac{H_{series}}{H_{parallel}} = \frac{R_p}{R_s} = \frac{R/2}{2R} = \frac{1}{4}\]

Q29. Establish a relation between current and drift velocity.

Ans. Consider a conductor of uniform cross-section area A, carrying a current I. Consider a small section KL of the conductor having a length $\Delta x$ or having a volume $ A.\Delta x$, then number of free electrons present in this section = $n A\Delta x$, where n = Number density of free electrons.

Total charge carried by these electrons while crossing the given section $\Delta Q = nAe\Delta x$

Now total time taken by the electrons to cross this section is $ \Delta t = \frac{\Delta x}{v_d} $Where $v_d$ = drift velocity of electrons

By definition \[I=\frac{\Delta Q}{\Delta t} = \frac{nAe\Delta x}{\Delta t} = neAv_d\]

Q30. Derive an expression for the current density of a conductor in terms of the drift speed of electrons.

Or

Prove that the current density of a metallic conductor is directly proportional to the drift speed of electrons.

Ans.  Current density \[J = \frac{I}{A} = \frac{neAv_d}{A} = nev_d\] Thus $ J \propto v_d $.

Q31. Derive an expression for drift velocity of free electrons in a conductor in terms of relaxation time.

Ans. We know that in the absence of an external electric field E, the conduction electrons in a conductor move randomly with velocities $ u_1, u_2, u_3, ….u_n$ such that their mean value \[\frac{u_1 +u_2+u_3+.....+u_n}{n} =0\]

However, in the presence of an external field E, electrons experience an acceleration \[\vec{a} = -\frac{e \vec{E}}{m}\]If $ t_1, t_2, t_3,…. $ be the times before two successive collisions for different electrons, then the final velocities acquired by different electrons are\[\vec{v_1} = \vec{u_1}+\vec{a}t_1, \vec{v_2} = \vec{u_2}+\vec{a}t_1, ...... \vec{v_n} = \vec{u_n}+\vec{a}t_n\]

                            

Mean value of electron velocity in the presence of an electrical field = Drift velocity $\vec{v_d} $ 

\[\frac{\vec{v_1}+\vec{v_2}+....+\vec{v_n}}{n} = \frac{\vec{u_1}+\vec{u_2}+...+\vec{u_n}}{n} + \vec{a}(\frac{t_1+t_2+.....+t_n}{n})\]

\[\vec{v_d} = \vec{a}\tau = -\frac{e\vec{E}}{m}\tau\]

Where relaxation time.\[\tau = \frac{t_1+t_2+...+t_n}{n}\]

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