Kinematics

1. The branch of Physics which deal with the study of motion of material objects is called Mechanics. Mechanics can be broadly classified into following branches :-

(i) Statics: It is a branch of mechanics that deals with the study of material objects at rest. 

(ii) Kinematics: It is that branch  of mechanics which deals with the study of the motion of material objects without taking into account the factors (i.e., nature of forces, nature of bodies, etc.) 

(iii) Dynamics: It is that branch of mechanics which deals with the study of motion of objects taking into account the factors which cause motion.                  

2. Rest: An object is said to be at rest if it does not change its position with time, with respect to its surroundings/observer.

3. Motion: An object is said to be in motion if it changes its position with time, with respect to its surroundings/observer.

4. Types of motion: 

(i) Rectilinear or translatory motion: Rectilinear motion is that motion in which a body, which is not a point mass body is moving along a straight line. Translatory motion is that motion in which a body, which is not a point mass body is moving such that all its constituent particles move simultaneously along parallel straight lines and shift through equal distance in a given interval of time. 

(ii) Circular or Rotatory motion: A circular motion is that motion in which a particle or a point mass body is moving on a circle. A rotatory motion is that motion in which a body, which is not a point mass body, is moving such that all its constituent particles move simultaneously along concentric circles, whose centers lie on a line, called the axis of rotation and shift through equal angle in a given time.

(iii) Oscillatory or Vibratory motion: Oscillatory motion is that motion in which a body moves to and fro or back and forth repeatedly about a fixed point (called mean position) in a definite interval of time. If in the oscillatory motion, the amplitude is very small, i.e., microscopic, the motion of body is said to be a vibratory motion.

5. Point mass object: An object can be considered as a point object if, during motion in a given time, it covers distances much greater than its own size.

6. Frame of reference: The frame of reference is a system of coordinate axes attached to an observer having a clock with him, with respect to which, the observer can describe position, displacement, acceleration etc. of a moving object. Inertial frame of reference is one in which Newton’s first law* of motion holds good. The non-inertial frame of reference is one in which Newton’s first law of motion does not hold good.

7. One dimensional motion: The motion of an object is said to be one-dimensional motion if only one out of the three coordinates specifying the position of the object changes with respect to time. For example, the motion of a train along a straight railway track, an object dropped from a certain height above the ground, a man walking on a level and narrow road, oscillations of a mass suspended from a vertical spring etc. belong to one-dimensional motion.

8. Two-dimensional motion: The motion of an object is said to be dimensional motion if two out of the three coordinates specifying the position of the object change with respect to time. For example, an insect crawling over the floor.

9. Three-dimensional motion: The motion of an object is said to be three-dimensional motion if all the three coordinates specifying the position of the object change with respect to time. For example : a kite flying on a windy day, the random motion of a gas molecule, a flying airplane or bird etc. belong to this type of motion.

10. The point followed by a point object during its called trajectory.

11. Scalar quantities or scalars: The physical quantities which have only magnitude but no direction, are called scalar quantities or scalars. For example, distance, length, work, charge, current, speed etc

12. Vector quantities or vectors: The physical quantities which have magnitude as well as direction are called vector quantities or vectors. Geometrically or graphically, a vector is represented by a straight line with an arrowhead, i.e. arrowed line.

13. Path length and displacement: The path length of an object in motion in a given time is the length of actual path traversed by a object in the given time. The displacement of an object in a motion of a given time is defined as the change in position of the object, i.e., the difference between the final and initial positions of the object of a given time. It is the shortest distance between the two positions of the object and its direction is from initial to final position of the object, during the given interval of time. It is represented by the vector drawn from the initial position to its final position. Path length is scalar and displacement is a vector quantity. Both are measured in meter (m).

14.  Speed: Speed of an object in motion is defined as a ratio of total path length (i.e., actual distance covered ) and the corresponding time taken by the object, i.e., 

             Speed = (total path length) / (time taken)    

15. Uniform speed: An object is said to be moving with a uniform speed, if it covers equal distances in equal intervals of time, howsoever small these intervals may be.

16. Variable speed: An object is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time, howsoever small these intervals may be.

17. Average speed: When an object is moving with a variable speed, then the average speed of the object is that constant speed with which the object covers the same distance in a given time as it does while moving with variable speed during the given time. Average speed for the given motion is defined as the ratio of the total distance traveled by the object to the total time taken i.e.,

                                             Average speed  =  (total distance traveled) / (total time taken)

If a particle travels distances S₁,S₂,S₃ etc. with speed v₁,v₂, v_3, etc. respectively, in same direction then total distance travelled = S₁ + S₂ + S₃ +…….  

Total time taken, \[t = \frac{S_1}{v_1}+\frac{S_2}{v_2}+\frac{S_3}{v_3}+.......\]

Total distance, \[S = S_1 + S_2 + S_3 +.....\]

Average velocity, \[V_{av} =\frac{S_1 + S_2 + S_3 +.....}{\frac{S_1}{v_1}+\frac{S_2}{v_2}+\frac{S_3}{v_3}+....... }\]

18. Instantaneous speed of an object at an instant of time t is defined as the limit of the average speed as the time interval () at the given instant of time, becomes infinitesimally small.                                    Instantaneous speed, \[\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}\]

19. Velocity of an object in motion is defined as the ratio of displacement and the corresponding time interval taken by the object, i.e., Velocity = (displacement)  / (time interval). 

Uniform velocity is that velocity of an object with which, it undergoes equal displacements in equal intervals of time howsoever small these intervals may be.

20. Graph: 

If an object is at rest, then the position-time graph is a straight line parallel to the time axis. see figure

If the object is in uniform motion along a straight line, starting from origin O, then the position-time graph is straight-line inclined to the time axis.   see figure

If an object is moving with constant negative velocity starting from a positive position then the position-time graph is a straight line. see figure

If the object is in non-uniform motion along a straight line then the position-time graph is a curve other than a straight line.    see figure

21. Displacement of the body is equal to the area of the velocity-time graph, during a given interval of time which is added with the proper sign. The area above the time axis is taken as positive while below is taken as negative in the velocity-time graph. 

22. If the body is moving along the straight line then the magnitude of the velocity and speed is equal and distance and displacement are also same. In uniform motion, the slope of the line gives the velocity of the object. Speed can never be negative.

23. The slope of velocity-time graph gives the acceleration.

24. It two bodies are moving with unequal velocities, their position-time graph must intersect each other.

25. Relative velocity of object B w.r.t. object A, \[\vec{v_{AB}} = \vec{v_B} - \vec{v_A}\]

26. Acceleration of an object in motion is defined as the ratio of change in velocity and the corresponding time taken by the object, i.e., Acceleration = (change in velocity) / (time taken)

27. Uniform acceleration. An object is said to be moving with a uniform acceleration if its velocity changes by equal amounts in equal intervals of time.

28. Variable acceleration. An object is said to be moving with a variable acceleration of its velocity changes by unequal amounts in equal intervals of time.

29. Average acceleration. When an object is moving with variable acceleration, then the average acceleration of the object for the given motion is defined as a ratio of the total change in velocity of the object during motion to the total time taken i.e., \[\vec{a_{av}} = \frac{\Delta \vec{v} }{\Delta t}\]

The slope of straight line joining two points on the velocity-time graph gives the average acceleration of the object between these two points. The average acceleration can be positive or negative depending upon the sign of the slope of the velocity-time graph. It is zero if the change in velocity of the object in the given interval of time is zero.

30. Instantaneous acceleration. When an object is moving with variable acceleration, then the object possess different acceleration at different instants.  \[\vec{a} = \lim_{\Delta t \to 0} (\frac{\Delta \vec{v} }{\Delta t}) = \frac{d\vec{v}}{dt}\]Instantaneous acceleration is also defined as the tangent to the velocity time graph at a position, corresponding to given instant of time.

31. The velocity-time graph for the motion for uniform velocity is parallel to the time axis. Velocity time graph of the accelerated motion is straight-line inclined with x-axis.

32. Equation of motion: \[v = u + at\] \[s = ut + \frac{1}{2}at^2\] \[v^2 = u^2 + 2as\]

where u is initial velocity, v is final velocity, a is acceleration, t is time, and s is displacement.

33. Distance travelled in nth second of uniformly accelerated motion is given by\[D_n = u + \frac{a}{2}(2n-1)\]

34. For motion under free fall, the equations of motion will be modified as \[v = gt\] \[h = \frac{1}{2}gt^2\] \[v^2 = 2gh\] assuming initial velocity is zero.

35. Relation time is the time which a person takes to observe, think, and act.

36. Acceleration of the body is given by the first derivative of velocity and second derivative of the position. Velocity is the first derivative of the position. Distance/Displacement is given by integration of the velocity with time. While velocity is given by integration of the acceleration.

 37. Polar vectors. These are those vectors that have a starting point or a point of application.

38. Axial vectors. These are those vectors that represent the rotational effect act along the axis of rotation in accordance with right-hand screw rule.

39. Modulus of a vector. The magnitude of a vector is called the modulus of that vector. The magnitude of vector $ \vec{A} $ is given by | A |. 

40. Unit vector a unit vector is the given vector is a vector of unit magnitude and has the same direction as that of the given vector. A unit vector in a given direction is also defined as a vector in that direction divided by the magnitude of the given vector. It is unitless and dimensionless vector and represents direction only.     \[\hat{A} = \frac{\vec{A}}{|A|}\]

41. Equal vectors. Two vector are said to be equal magnitude and same direction.

42. Negative vector. A negative vector of a given vector is a vector of same magnitude but acting in a direction opposite to that of the given vector.

43. Co-initial vectors. The vectors are said to be co-initial if their initial point is common.

44. Collinear vectors. These are those vectors which are having equal or unequal magnitudes and are acting along the parallel straight lines.

45. Coplanar vectors. These are those vectors which are acting in the same plane.

46. The multiplication of a vector A by a real number n becomes another vector n A. its magnitude becomes n times the magnitude of the given vector. Its direction is the same or opposite as that of A, according as n is a positive or negative real number. When a vector A is multiplied by a scalar S, it becomes a vector S A, whose magnitude is S times the magnitude of A and it acts along the direction of A . the unit of S A, is different from the unit of vector A. 

46. Resultant vector of two or more vectors is defined as that single vector which produced the same effect as is produced by individual vectors together.

47. Vector addition, It is state that the vectors to be added are arranged in such a way so that the head of first vector coincides with the tail of second vector, whose head coincides with the tail of third vector and so on, then the single vector drawn from the tail of the first vector to the head of first vector represent their resultants vector.

48. Triangle law of vectors states that if two vectors acting on a particle at the same time are represented in magnitude and direction by two sides of a triangle taken in one order, their resultant vector is represented in magnitude and directed by the third side of the triangle taken in the opposite order. Let two vectors $ \vec{A} $ and $ \vec{B} $ acting at an angle $ \theta $ is given by \[R = \sqrt{A^2 + B^2 + 2ABcos\theta }\] Direction of resultant vector with vector A is given by \[tan\alpha = \frac{Bsin\theta}{A + Bcos\theta}\]

49. Parallelogram law  of vector s state that if two vectors acting on a particle at the two adjacent side of a parallelogram drawn from a point, their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point. Resultant of two vector is same as given by triangle law of vector addition.

50. Polygon law of vectors states that if any number of vectors, acting on a particle at the same time are represented in magnitude and direction by various sides of an open polygon taken in the same order, their resultant is represented in magnitude and direction by the closing side of the polygon taken in opposite order.

51. Lami’s theorem. It states that if three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. i.e., \[\frac{A}{sin\alpha} = \frac{B}{sin\beta} = \frac{C}{sin\gamma}\]

52. The object is in equilibrium if there is no linear motion of the object i.e., the resultant force on the object is zero, There is no rotational motion of the object i.e., the torque due to forces on the object is zero and There is minimum potential energy of the object for stable equilibrium.

53. A single vector which balances two or more vectors acting on a body at the same time is called an equilibrant vector.

54. It is that vector which is zero magnitude and an arbitrary direction. A zero vector is represented by 0 (arrow over the number zero). It is also called null vector. When a vector is multiplied by zero, the result is a zero vector. i.e. 0 (A) = 0. The result of  addition of a vector to its own negative vector is a zero vector, i.e. A + (-A) =0.

55. Resolution of a vector, It is the process of splitting a single vector into two or more vectors in different directions which together produce the same effect as it produced by the single vector alone. The vectors into which the given single vector is split are called component vectors.

56. Uniqueness of component vectors. It is to be noted that the resolution of vector R into two component vectors along the direction of vector A and B is unique. 

57. A body is said to be projectile if it has motion in two dimensions i.e. a projectile should have two-component velocities in two mutually perpendicular directions.  

58. For horizontal projectile, Equation of trajectory is given by \[y = \frac{g}{2u^2}x^2\]Time of flight is given by, \[T = \sqrt{\frac{2h}{g}}\]Horizontal Range, i.e. distance covered along horizontal axis is given by,\[x = u\sqrt{\frac{2h}{g}}\]Velocity of object at any time is given by,\[v = \sqrt{u^2 + g^2t^2}, tan\beta = \frac{gt}{u}\]

59. For angular projectile, Equation of the trajectory is given  by, \[y = xtan\theta - (\frac{1}{2}\frac{g}{u^2cos^2\theta})x^2\] Time of flight is given by,\[T = \frac{2usin\theta}{g}\]Maximum height obtained by the projectile is, \[H = \frac{u^2sin^2\theta}{2g}\]Range of the projectile is given by,\[R = \frac{u^2sin2\theta}{g}\] where $ \theta $ is angle of projection.

60. Horizontal range of projectile is same when angle of projection is (i) $ \theta $ and 90⁰ – $ \theta $ or (ii) (45⁰ + $ \theta $) and (45 degree – $ \theta $).

61. At the higher point, the projectile possesses velocity only along horizontal direction. At the highest point of the projectile path, the velocity and acceleration are perpendicular to each other.

61. In projectile motion, the particle return to the ground at the same angle and with the same speed with which it was projected. In projectile motion, kinetic energy is maximum at the point of projection or point of reaching the ground and is minimum at the highest point. There are two times for which the projectile travels the same vertical distance and the sum of these timings is equal to the total time of flight of projectile. 

62. The time of flight, the horizontal range and max. height are independent of mass of projectile. The maximum height attained by projectile is equal to one fourth of its maximum range.

63. Angular displacement of the object moving around a circular path is defined as the angle traced out by the radius vector at the center of the circular path in a given time. It is denoted by $ \theta $.

64. Angular velocity, of an object in circular motion is define das the time rate of change of its angular displacement. i.e. \[\omega = \frac{d \theta }{d t}\]

65. Angular acceleration, of an object in circular motion is defined as the time rate of change of its angular velocity. i.e. \[\alpha = \frac{d \omega }{d t} = \frac{d^2 \theta }{d t^2}\]

66. Uniform circular motion,  When a point object is moving on a circular path with a constant speed (i.e. it covers equal distance of the circumference of the circle in equal intervals of time), then the motion of the object is said to be a uniform circular motion Time period in circular motion, is defined as the time taken by the object to complete one revolution on its circular path. Frequency in circular motion is defined as the number of revolutions completed by the object on its circular path in a unit time.

67. Relation between the time period and frequency, $ \nu T = 1 $

68. Relation between angular velocity, frequency and time period $ \omega = \frac{\theta}{t} = \frac{2\pi}{T} = 2\pi \nu $

69. Centripetal acceleration,  Acceleration acting on the object undergoing circular motion is called centripetal acceleration. When a body is moving with a constant angular velocity, its angular acceleration is zero. It is given  by, \[|\vec{a}| = \omega^2 r = \frac{v^2}{r}\]

70. Uniform circular motion is an accelerated motion. The work done by the centripetal force is zero. The centripetal force does not increase the kinetic energy and angular momentum of the particle moving in a circular path.

71. Without centripetal force, a body cannot move on a circular path. In different types of circular motion, the centripetal force is provided by different means.

72. Total acceleration of the body moving in circle is given by,  \[\vec{a} = \vec{a_c}+\vec{a_T}\]

\[|\vec{a}| = \sqrt{(a_c^2 + a_T^2)}\]

Video Lecture:

1 Kinematics, Frame of reference, scalar and vectors  watch video

2 Distance and displacement watch video

3 Speed and velocity watch video

4 Acceleration and equation of motion by graphical method watch video

5 Questions based on graphical method watch video

6 Equations by calculus method watch video 

7 Vector definitions and triangle law of vector addition watch video

8 Zero vector, lami's theorem and vector resolution watch video

9 Dot and cross product of two vectors watch video

10 Distance covered in n-th second and motion in vertical direction watch video

11 Relative velocity 1 watch video

12 Relative velocity 2 watch video

13 Horizontal projectile watch video

14 Angular projectile watch video

15 Circular motion watch video 

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Electrostatics (Practice questions)

1. A negatively charged ebonite rod attracts a suspended ball of straw. Can we infer that the ball is positively charged? [No]

2. Can two similarly charged balls attract each other? [Yes]

3. How can you charged a metal sphere negatively without touching it? [Induction]

4. If two objects repel one another, you know both carry either a positive charge or negative charge. How would you determine whether these charges are positive or negative? [Repulsion Test]

5. Does motion of the body affect its charge? [No]

6. What is the dimensional formula for $ \epsilon_0 $? [$M^{-1}L^{-3}T^3A^2$]

7. Two small balls having equal positive charge q coulomb are suspended by two insulating string of equal length l meter from a hook fixed to a stand. The whole setup is taken in a satellite into space where there is no gravity. What is the angle between the two strings and the tension in each string? [$180^0$]

8. Two point charges of + 2μ C and + 6 μ C  repel each other with a force of 12 N. If each is given an additional charge of -4μ C, what will be the new force? [$-4N$]

9. Two point charges of 10^-8‑C and -10^-8 C are placed 0.1 m apart. Calculate electric field intensity at A, B, and C shown in figure. [$E_A = 7.2 \times 10^4 N/C $ along AQ, $E_B = 3.2 \times 10^4 N/C $ along PB, $E_C =9 \times 10^3 N/C $ parallel to PQ]

10. When does a charged circular loop behave at a point charge? [When the point is very very far away ]

11. How does a free electron at rest move in an electric field? [Opposite to Electric Field]

12. What does (q₁ + q ₂) = 0 signify? [Dipole]

13. Two-point charges of +16 μ C and -9 μ C are placed 8 cm apart in the air. Determine the position of the point at which the resultant electric field is zero. [24 cm to the right of -9$\mu C$]

14. Four particles, each having a charge q are placed on the four corners A, B, C, D of a regular pentagon ABCD. The distance of each corner from the center is a. Find the electric field at the center of the pentagon. [$\frac{q}{4\pi \epsilon_0 a^2}$ along OE]

15. Two charges of -4 μ C and + 4 μ C are placed at the points A (1, 0, 4) and B (2, -1, 5) located in an electric field E = 0.20 $\hat{i}$ V/cm. Calculate the torque acting on the dipole. [$1.131 \times 10^{-4} N-m $]

16. Can we produce high voltage on the human body without getting a  shock? [Yes]

17. Do electron tend to go to region of high potential or low potential? [High Potential]

18. In a certain 0.1 m³ of space, electric potential is found to be 5 V throughout. What is the electric field in this region? [$E = 0$]

19. Write an expression for potential the energy of two charges  q₁ and q₂ at r₁ and r₂ in a uniform electric field E. [$P.E. = q_1V(\vec{r_1})+q_2V(\vec{r_2})+ \frac{q_1q_2}{4\pi \epsilon_0 |\vec{r_1}-\vec{r_2}|}$]

20. Two point charges 4 μ C and -2 μ C are separated by a distance of 1 m in air. Calculate at what point on the line joining the two charges in the electric potential zero? [$\frac{2}{3}m $ from $4\mu C$ ]

21. An electric field of  20 N/C exists along the X-axis in space. Calculate the potential difference (V_B – V_A) where the coordinates of A and B are given by (i) A (0, 0); B (4m, 2 m) (ii) A (4 m, 2m); B (6 m, 5 m). [$-80V,-40V$]

22. If the potential in the region of space around the point (-1 m, 2m, 3m) is given by V = (10 x² + 5 y² – 3 z²), calculate the three component of electric field at this point. [$E_x=20Vm^{-1},E_y=-20Vm^{-1},E_z=18Vm^{-1}$]

23. The electric field in a certain region of space is $(5\hat{i} + 4\hat{j} -4 \hat{k})$ x 10⁵ N/C. calculate electric flux due to this field over an area of $ (2\hat{i} – \hat{j})$ x 10^-2 m². [$6 \times 10^3 NC^{-1}m^2$]

24. A point charge q moves from a point P to the point S along the path PQRS in a uniform electric field E along the positive direction of the x-axis. Calculate work done in this process, when co-ordinate of P, Q, R,S are (a, b, 0), (2a 0, 0), (a, -b , 0) and (0, 0, 0) respectively. [$-qEa$]

25. Find the capacitance of the combination shown in figure between A and B. [$1\mu F$]

26. A network of four 10 μ F capacitors is connected to a 500 V supply, as shown in figure. Determine the (a) equivalent capacitance of the network and (b) charge on each capacitor. [$C=13.3\mu F, Q_1=Q_2=Q_3=1.7 \times 10^{-3}C, Q_4=5\times 10^{-3}C$ ]

27. Find equivalent capacity between A & B, as shown in figure [$1\mu F$]

28. In figure, find equivalent capacity between A and B. [$5\mu F$]

29. A slab of material of dielectric constant K has the same area as the plates of a parallel plate capacitor but has a thickness (3/4)d, where d is the separation of the plates. How is the capacitance changed when the slab is interested between the plates. [$C=\frac{4K}{3+K}C_0$]

30. Two spheres of radii R and 2 R are charged so that both of these have the same surface charge density. The spheres are located away from each other and are connected by a thin conducting wire. Find the new charge density on the two spheres. [$\sigma_1' = \frac{5}{3}\sigma,\sigma_2' = \frac{5}{6}\sigma$]

31. A spherical shell of radius b with charge Q is expended to radius a. Find the work done by the electric force in the process? [$W = \frac{Q^2}{8\pi \epsilon_0}[\frac{1}{a}-\frac{1}{b}]$]

32. Sketch a graph to show how charge Q is given a capacitor of capacity C varies with the potential difference V. [ Figure ] 

33. The space between the plate of a parallel plate capacitor is filled consecutively with two dielectric layers of thickness d₁ and d₂ having relative permittivities $\epsilon_1$ and $\epsilon_2$ respectively. If a is area of each plate, what is the capacity of a capacitor? [$C = \frac{\epsilon_0 A}{\frac{d_1}{\epsilon_1}+\frac{d_2}{\epsilon_2}}$]

34. The equivalent capacitance of the combination between A and B in the given figure is 4 μ F. pageno.1/160 (i) Calculate the capacitance of capacitor C. (ii) Calculate charge on each capacitor if 12 V battery is connected between A and B. (iii) Calculater potential drop across each capacitor. [$5\mu F, 48\mu C, 2.4V,9.6V$]

35. Calculate the capacitance of the capacitor C in the figure. The equivalent capacitance of the combination between P and Q is 30 μ F. [$60\mu F$]

36. A combination of four identical capacitors is shown in figure . If resultant capacitance of the combination between the point A and D is 1 μ F. Calculate capacitance of each capacitor. [$4 \mu F$]

37. A parallel plate capacitor is filled with a dielectric as shown in figure. What is its capacitance? [$\frac{2\epsilon_0 AK_1K_2}{d(K_1+K_2)}$]

38. Three capacitors of capacitances 2 μ F, 3 μ F and 6 μ F are connected in series with a 12 V battery. All the connecting wires are disconnected. The three positive plates are connected together and the three negative plates are connected together. Find the charges on the three capacitors after the reconnection. [$\frac{72}{11}\mu C,\frac{108}{11}\mu C,\frac{216}{11}\mu C$]

38. Calculate the charges which will flow in sections 1 and 2 in figure, when key K is pressed. [$EC_1,\frac{EC_1C_2}{(C_1+C_2)}$]

39. In the circuit shown in figure, the emf of each battery is E = 12 volt and the capacitance are C₁ = 2.0 μ F and C₂ = 3.0 μ F. Find the charges which flow along the paths 1, 2, 3 when key K is pressed. [$24\mu C,-36\mu C,12\mu C$] 

40. Calculate the equivalent capacitance between the point A and B in the combination shown in figure [$13.44 \mu C$]

41. If C₁ = 3 pF and C₂ = 2 Pf, calculate the equivalent capacitance of the network shown in figure between points A and B. [$1pF$]

42. Find the equivalent capacitance of the combination of capacitors between the points A and B as shown in figure. Also, calculate the total charge that flows in the circuit, when a 100 V battery is connected between the points A and B. [$C = 20\mu F, Q=2\times 10^{-3}C$]

43. A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure The width of each stair is a and the height is b. Find the capacitance of the assembly. [$C = \frac{\epsilon_0 A(3d^2+6bd+2b^2)}{3d(b+d)(d+2b)}$]

44. Find out the potential difference across the plates of 1 μ F capacitor in figure. [$3.82 V$]

45. Find the capacitance of three parallel plates, each of area A m² and separated by d₁ and d₂ meter. The in-between spaces are filled with dielectrics of relative permittivity $\epsilon_1$ and $\epsilon_2$. The permittivity of free space in $\epsilon_0$. [$C = \frac{\epsilon_1 \epsilon_2 \epsilon_0 A}{\epsilon_1d_2 + \epsilon_2 d_1}$]

46. An uncharged capacitor is connected to a battery. Show that half the energy supplied by the battery is lost as heat while charging the capacitor. 

47. Obtain the formula for the electric field due to a long thin wire of uniform linear charge density λ without using Gauss’s law. [$E = \frac{\lambda}{2\pi \epsilon_0 r}$]

48. A particle of mass m and charge (-q) enters the region between the two charged plates initially moving along x-axis with speed v_x,  the length of plate is l and a uniform electric field E is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is \[\frac{qEL^2}{(2 m v_x^2)}\].Compare this motion with the motion of a projectile in a gravitational field. 

49. A spherical conducting shell of inner radius r₁ and outer radius r₂ has a charge Q. (a)  A charge q is placed at the center of the shell. What is the surface charge density on the inner and outer surfaces of the shell? (b)  Is the electric field intensity inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.[(i) $\sigma_1 = -\frac{q}{4\pi r_1^2},\sigma_2 = \frac{Q+q}{4\pi r_2^2}$, (ii) $Yes$]

50. Two charges q and -3q are placed fixed on x-axis separated by distance ‘d’. Where should a third charge 2q be placed such that it will not experience any force? [$x = \frac{(1+\sqrt{3})d}{2}$ from 2q]

51. In 1959, Lyttleton and Bondi suggested that the expansion of the universe could be explained if matter carried a net charge. Suppose that the universe is made up of hydrogen atoms with a number density N, which is maintained a constant. Let the charge on the proton be: e_p = -(1 +y) e where e is the electronic charge. (a)    Find the critical value of y such that expansion may start. (b)    Show that the velocity of expansion is proportional to the distance from the centre. [(a) $\approx 10^{-18}$]

52. Consider a sphere of radius R with charge density distributed as $ \rho (r) = kr $ for r< R,  =0 for r> R. (a)    Find the electric field at all points r. (b)  Suppose the total charge on the sphere is 2e, where e is the electron charge. Where can two proton be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution. [(a) For r<R, $E = \frac{kr^2}{4\epsilon_0}$, For r>R, $E = \frac{kR^4}{4\epsilon_0 r^2}$ (b) $r = \frac{R}{8^{1/4}}$ from center of sphere]

53. Two fixed, identical conducting plates ($ \alpha $ and $\beta $), each of surface area S are charged to -Q and q, respectively, where Q > q > 0. A third identical plate $ \gamma $, free to move is located on the other side of the plate with charge q at a distance d as shown in figure. Third plate is released and collides with the plate $\beta $. Assume the collision is elastic and the time of collision is sufficient to redistributed charge amongst $\beta $ and $\gamma $. (a)    Find the electric field acting on the plate $\gamma $ before collision. (b)   Find the charges on $\gamma $ and $\beta $ after the collision. (c)  Find the velocity of the plates $\gamma $ after the collision and at a distance d from the plate $\beta $. [(a) $E = \frac{q-Q}{2\epsilon_0 S}$, (b) $q_{\beta} = (Q+\frac{q}{2}),q_{\gamma} = \frac{q}{2}$, (c) $(Q-\frac{q}{2})\sqrt{\frac{d}{m\epsilon_0 S}}$]

54. There is another useful system of units, besides the SI/mks A system, called the cgs (centimeter-gram-second). In this system, Coloumb’s law is given by \[F = \frac{Qq}{r^2} \hat{r}\]where the distance r is measured in cm (= 10^-2m), F in dynes (= 10^-5 N ) and the charges in electrostatic units (es units), where 1 es unit of charge = $ \frac{1}{[3]} $ x 10^-9 C. The number [3] actually arises from the speed of light in vacuum which is now taken to be exactly given by c = 2.99792458 x 10⁸ m/s.An approximate value of c then is c =[3] x 10⁸   m/s. Show that the coulomb law in cgs  unit yield 1 esu of charge = 1 (dyne)^1/2 cm. Obtain the  dimensions  of units of charge in terms of mass M, lengh L and time T. Show that it is given in terms of fractional powers of M and L. Write 1 esu of charge = xC, where x is a dimenionless number. Show that this gives \[\frac{1}{4\pi \epsilon_0} = \frac{10^{-9}}{x^2} N.m^2/C^2\]    

55. Two charges -q each are fixed separated by distance 2d. A third charge q of mass m placed at the mid-point is displaced slightly by x (x << d) perpendicular to the line joining the two fixed charges as shown in figure. Show that q will perform simple harmonic oscillation of time period. \[T= [\frac{8\pi^3 \epsilon_0 md^3}{q^2}]^{1/2}\]

56. Total charge -Q is uniformly spread along length of a ring of radius R. A small test charge +q of mass m is kept at the centre of the ring and is given a gentle push along the axis of the ring. Show that the particle executes a simple harmonic oscillation. Obtain its time period. [(b) $T = 2\pi \sqrt{\frac{4\pi \epsilon_0 mR^3}{Qq}}$]

57. Prove that a closed equipotential surface with no charge within itself must enclose an equipotential volume.

58. Calculate potential energy of a point charge -q placed along the axis due to a charge +Q uniformly distributed along a ring of radius R. Sketch P.E. as a function of axial distance z from the centre of the ring. Looking at graph, can you see what would happen if -q is displaced slightly from the centre of the ring (along the axis)? [$U = \frac{-qQ}{4\pi \epsilon_0\sqrt{R^2+z^2}}$]

59. Find the equation of the equipotential for an infinite cylinder of radius r₀, carrying charge of linear density $ \lambda $. [$r=r_0e^{-2\pi \epsilon_0[V(r)-V(r_0)]/\lambda}$]

60. Two point charges of magnitude + q and -q are placed at (- d/2, 0, 0) and (d/2, 0, 0), respectively. Find the equation of the equipotential surface where the potential is zero. [$x=0$]

61. A parallel plate capacitor is filled by a dielectric whose relative permittivity varies with the applied voltage (u) as $ \epsilon  =\alpha U $ where $\alpha $ = 2 V^-1. A similar capacitor with no dielectric is charged to U₀ = 78 V. It is then connected to the uncharged capacitor with the dielectric. Find the final voltage on the capacitors. [$6V$]

62. A capacitor is made of two circular plates of radius R each, separated by a distance d <<R. The capacitor is connected to a constant voltage. A thin conducting disc of radius r << R and thickness t << r is placed at the centre of the bottom plate. Find the minimum voltage required to lift the disc if the mass of the disc is m. [$V = \sqrt{\frac{mbd^2}{\pi \epsilon_0 r^2}}$]

63. In a circuit shown in figure, initially K₁ is closed and K₂ is open. What are the charges on each capacitor. Then K₁ was opened and K₂ are closed (order is important ). What will be the charge on each capacitor now?[ C = 9$\mu $F].  

64. Calculate potential on the axis of a disc of radius R due to a charge Q uniformly distributed on its surface. [$V = \frac{2Q}{4\pi \epsilon_0 R^2}[\sqrt{R^2+z^2}-z]$]

65. Two charges q₁ and q₂ are placed at (0, 0, d) and (0, 0, -d) respectively. Find locus of points where the potential a zero. [$x^2+y^2+z^2+[\frac{(q_1/q_2)^2+1}{(q_1/q_2)^2-1}](2zd)+d^2 = 0$]

66. Two charges -q each are separated by distance 2d. A third charge +q is kept at mid point O. Find potential energy of +q as a function of small distance x from O due to -q charges. Sketch P.E. v/s x and convince yourself that the charge at O is in an unstable equilibrium.

67. Two point masses, m each carrying charge -q and +q are attached to the ends of a massless rigid non conducting rod of length l. The arrangement is placed in a uniform electric field E such that a rod makes a small angle  = 5⁰ with the field direction. Show that the minimum time needed by the rod to align itself along the field (after it is set free)is \[T =\frac{\pi}{2}\sqrt{\frac{ml}{2qE}}\] 

68. Plate A of a parallel plate air filled capacitor is connected to a spring having force constant k and plate B is fixed. They are held on a frictionless tabletop as shown in figure. If a charge +q is placed on plate A and a charge -q on plate B, how much does the spring expand? [$\frac{q^2}{2\epsilon_0 Ak}$]

69. Find the capacitance of the infinite ladder between points X and Y, as shown in figure [$2\mu F$]

70. Two identical charged sphere are suspended in air by strings of equal length and make an angle of 30⁰ with each other. When suspended in a liquid of density 0.8 g/cc., the angle remain the same. What is the dielectric constant of the liquid? Take density of the material of the sphere = 1.6 g/c.c. [$K=2$]

71. A thin fixed ring of radius 1 m has a positive charge of 10^-5 C uniformly distributed over it. A particle of mass 0.9 gram and having a negative charge of 10^-6 C is placed on the axis at a distance of 1 cm from the centre of the ring. Show that the motion of the negatively charged particle is approximately SHM. Calculate the time period of oscillation. [$T=0.628 s$]

72. Find the potential difference between the left and right plate of each capacitor in the circuit shown in  figure.  [$V_1 = \frac{(E_2-E_1)C_2}{C_1+C_2},V_2 = \frac{(E_2-E_1)C_1}{C_1+C_2}$]

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Measurement and Error

1 Radian (rad): One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. \[d\theta =\frac{ds}{r}\]SI unit is radian.

2. Steradian (sr): One steradian is the solid angle subtended at the center of a sphere, by that surface of the sphere, which is equal in area, to the square of the radius of the sphere. \[d\Omega = \frac{dA}{r^2}\]SI unit is steradian

3. Astronomical Unit (AU): It is the average distance of the centre of the sun from the centre of the earth. 1 AU =1.496 x 10¹¹ m =1.5 x 10¹¹ m

4. Light year (ly): One light year is the distance travelled by light in vacuum in one year. 1 ly = 9.46 x10¹⁵m.

5. Parsec : One parsec is the radius of the circle at the centre of which an arc of the circle, 1 AU long subtends an angle of 1”. 1 parsec = 3.1 x 10¹⁶ m

6. Relation between AU, ly and par sec, 1 ly = 6.3 x 10⁴ AU, 1 par sec = 3.26 ly

7. In the micro-cosm measurement,

(i)                  1 micron = 1 μ or 1 μm = 10^-6 m

(ii)                1 nanometer = 1 nm =10^-9 m

(iii)               1 angstrom = 1A⁰ = 10^-10 m

(iv)               1 fermi = 1 femtometer = 1 fm = 10^-15 m

8. For measuring very small area,

1 acre = 4047 m²

1 are (a) = 10² m²

1 hactare = 10⁴ m²

9. For measuring heavy masses,

(i)                  1 tonne or  1 metric ton = 1000 kg

(ii)                1 quintal = 100  kg

(iii)               1 slug = 14.57 kg

(iv)               1 lb = 0.4536 kg

10. For measuring very small masses, 1 atomic mass unit = 1 a.m.u. or 1 u = 1.66 x 10^-27 kg.

11. Some practical units of standard of time are :

(i)Solar day: It is the time interval between two successive passage of the sun across the meridian.

(ii) Sedrial day: It is the time interval between two successive passages of a fixed star across the meridian. 

(iii) Solar year (or year) is the time taken by the earth to complete one revolution around the sun in its orbit. 

1 solar year = 365.25 average solar days = 366.25 sedrial days

The year in which there is total solar eclipse is called a tropical year. The year which is divisible by 4, and I which month of February has 29 days, is called a leap year. One hundred years make up 1 century. 

(iv) Lunar month. It is the time taken by moon to complete one revolution around the earth in its orbit. 1 Lunar month = 27.3 days. 

(v) Shake: It s the smallest practical unit of time. 1 shake = 10⁸s

12. Parallax method: Parallax is the name of the name given to change in the position of an object with respect to the background when the object is seen from two different positions. The distance between the two-position (i.e., points of observation) is called the basis. \[\Theta = \frac{b}{x}\] where b is the arc length, x is the radius and $ \Theta $ is angle subtended. The parallax method has been used for measuring the distance of stars of which are less than a hundred light-years away.

13. Error: The difference between the true value and the measured value of any physical quantity is called error. i.e. Error = [True Value] - [Measured Value].

14. There are three types of errors, namely, Systematic, Random and gross error.

15. Systematic Error: There error tend to be in any one direction either positive or negative. some of the systematic errors are: Instrumental error, Imperfection in experimental technique or procedure, Personal error, Least count error.

16. Random Error: These errors occur irregularly. It arises due to random and unpredictable variation in experimental conditions like Temperature, Pressure, voltage, etc. It can be minimized by repeating the experiments. 

17. Gross Error: These errors arise due to the carelessness of the observer. For example, Reading an instrument improperly, noting observations incorrectly, using wrong values in the calculation, etc.    

18. Absolute Error: It is the magnitude of the difference between the true value and the individual measured value of the quantity. Let physical quantity be measured n times and observed values be $ a_1, a_2,........,a_n $. Then, arithmetic mean of these value are, \[a_m = \frac{a_1+a_2+.....+a_n}{n} \implies a_m =\frac{1}{n}\Sigma_{i=1}^{i=n}a_i\]Then absolute error in any measured value is given by \[\Delta a_i = a_m - a_i\]

19. Means absolute error: It is arithmetic mean of the magnitude of absolute errors in all measurements of quantity. It is represented by $\Delta a_{mean} $. Thus, \[\Delta a_{mean}=\frac{|\Delta a_1|+|\Delta a_2|+.....+|\Delta a_n|}{n} \implies \Delta a_{mean} = \frac{1}{n}\Sigma_{i=1}^{i=n}|\Delta a_i |\]Hence final result of measurement may be written as  

  \[a=a_m \pm \Delta a_{mean}\]

20. Relative Error: It is defined as the ratio of mean absolute error to the mean value of the quantity measured. Thus, \[\delta a = \frac{\Delta a_{mean}}{a_m}\]

21. Error in Sum: Let x = a + b, then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\]Hence maximum absolute error in sum of two quantities is equal to sum of the absolute errors in the individual quantities.

22. Error in difference: Let x = a - b,  then maximum absolute error in x is \[\Delta x = \pm (\Delta a +\Delta b)\] Hence maximum absolute error in difference of two quantities is equal to sum of the absolute errors in the individual quantities.

23. Error in Product: Let  $ x $ = a x b, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

24. Error in Product: Let  $ x = \frac{a}{b} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm (\frac{\Delta a}{a} +\frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

25. Error in case of measured quantity raised to a power: Let  $ x = \frac{a_n}{b_m} $, then maximum absolute error in x is \[\frac{\Delta x}{x} = \pm ( n \frac{\Delta a}{a} + m \frac{\Delta b}{b})\]Hence maximum absolute error in product is equal to sum of fractional or relative errors in individual quantities. 

Video Lecture:

Measurement and errors: Watch video

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Moving Charges and Magnetism

1. Earlier electricity and magnetism were considered two separate domains of Physics. However, on the basis of Oersted’s experiment and subsequent work it has been established that moving charges or currents produce a magnetic field in the surrounding space. Subsequently on the basis of more intense experimentation unified basic laws of electromagnetism were developed, which led to the discovery of electromagnetic waves.

2. Magnetic field is characterised by a magnetic field vector (also known as magnetic induction or magnetic flux density) $ \vec{B} $. SI unit of magnetic field $ \vec{B} $ is 1 tesla (1 T). Sometimes it is also referred as weber/m² (Wb m² ). C.G.S. unit of  $ \vec{B} $ is 1 gauss (1 G), where 1 G = 10^-4 T.

3. Due to a straight conductor, the magnetic field lines formed are concentric circles around the conductor with the conducting wire at the centre. The magnetic line formed closed loops with no beginning and no end.

4. The direction of magnetic field due to a straight, current carrying conductor is given by right hand thumb rule. According to this rule, grasp the conductor carrying current in your right hand with the thumb perpendicular to the fingers and pointing in the direction of the current, then the curls of the figures point in the direction of magnetic field $ \vec{B} $ associated with the conductor.

5. According to Biot-savart law, the magnetic field $\vec{dB} $ in free space at a point P at a distance ‘r’ from a differential current element $ I \vec{dl} $ is given by \[\vec{dB} = \frac{\mu_0}{4\pi} \frac{I\vec{dl}x \vec{r}}{r^3}\]Direction of $ \vec{dB} $ is that of  $ \vec{dl}* \vec{r}$ i.e., direction of magnetic field is perpendicular to both $\vec{dl} $ and $ \vec{r} $. Magnitude of magnetic field is given by \[dB = \frac{\mu_0}{4\pi} \frac{Idl sin \theta}{r^2}\]

6. The term μ₀ appearing in the Biot-Savart law is known as the magnetic permeability of free space. Value of μ₀ = 4π * 10^-7 tesla metre / ampere (T m A^-1).

7. For a straight current carrying thin wire of finite length the magnetic field at a point situated at a normal distance R from the centre of conductor is given by \[B = \frac{\mu_0 I}{4\pi R} [ sin \phi_1 + sin \phi_2]\] where I is the current flowing in the wire and $ \phi_1 $ and $ \phi_2 $ are the angles by the two ends of given conductor from normal direction. 

8. For a current carrying thin wire of infinite length $ \phi_1 = \phi_2  = 90 $ and hence \[B = \frac{\mu_0 I}{2\pi R}\]However, for a point P lying on the thin wire itself, the magnetic field B is zero. Magnetic field at a point situated near one end of an infinitely long current carrying wire is given by \[B =\frac{\mu_0 I}{4\pi R}\]

9. For a circular wire loop of radius R and carrying a current I the magnetic field at the centre point of circle is given by \[B = \frac{\mu_0I}{2R}\]Direction of magnetic field due to a circular loop carrying current is given by right hand palm rule. According to it, curl the palm of your right hand around the circular wire with the fingers pointing in the direction of the current. Then the thumb of right hand gives the direction of the magnetic field. If we have a circular coil of N turns then the magnetic field at its centre is given by \[B = \frac{\mu_0 N I}{2R}\] 

10. Magnetic field B due to a current I in a circular coil of N turns each of radius R at a point P on its axial line at a distance x from the centre of coil is given by \[B = \frac{\mu_0 N I R^2}{2 (R^2 + x^2 )^{3/2} }\]

The above expression reduces to the following form : 

(i) At the centre of circular coil (i.e., x = 0),  \[B = \frac{\mu_0N I}{ 2R }\] 

(ii) At points far away from the centre i.e., x >> R, \[B = \frac{\mu_0 N I R^2}{ 2x^3} = \frac{\mu_0}{4\pi}\frac{ 2 I A}{x^3}\] where  A = π R² = area of circular loop. The magnetic field is directed axially in the direction given by right hand rule. 

11. If an open surface is bounded by a loop then Ampere’s circuital law states that the line integral of the magnetic field along the loop is equal to μ₀ times the net current passing through the surface. By the net current we mean the algebraic sum of the current within that loop i.e., where I_e is the net current enclosed. While applying Ampere’s law we follow the right hand rule. Let the finger of the right hand be curled in the sense the boundary is traversed in the loop along the direction of integral, then the direction of the thumb gives the sense in which the current I is regarded as positive.

12. While applying Ampere’s circuital law, if we choose the amperian loop such that at each point of the loop either $ \vec{B} $ is tangential to the loop and is a non-zero constant B or $ \vec{B} $ is normal to the loop, then \[\oint \vec{B}.\vec{dl} = BL = \mu_0 I_e\]Here L is the length of the loop for which B is tangential.

13. The simplest application of Ampere’s circuital law is to determine magnetic field at a point situated at a normal distance R from a long current carrying straight wire of a circular cross-section. From this we find that  

(i)\[B = \frac{\mu_0 I}{2\pi R}\]where R >= r (r is the radius of the wire) and

(ii) \[B = \frac{\mu_0 I}{2\pi r^2}R\]when  R <= r

14. A straight solenoid is prepared by a long, insulted copper wire wound in the form of a helix with neighbouring turns very closely spaced. If the length of the solenoid is large enough as compared to its transverse cross-section, the solenoid is considered to be a long, straight solenoid. In a solenoid magnetic field due to all the turns is in the same direction and being added up. The net magnetic field inside a tightly wound infinite solenoid is uniform and axial but zero outside the solenoid. Magnetic field inside the infinitely, long solenoid carrying current I by applying Ampere's Circuital Law is given by \[B = \mu_0 nI = \frac{\mu_0 NI}{l}\]Where n = number of turns per unit length =$ \frac{N}{L} $. The direction of the magnetic field is given by right hand palm rule. Magnetic field due to an infinitely long solenoid is along the axis of the solenoid. At a point just near the free end of a long, straight solenoid the magnetic field \[B = \frac{μ0 n I}{2}\]

15. A toroid is a hollow circular ring on which a large number of turns of an insulted copper wire are closely wound. For a toroidal solenoid carrying current the magnetic field at any point : (i) outside the toroid, (ii) inside the open space in the toroid is zero. Magnetic field inside a toroidal solenoid of radius R and having N turns in all and carrying a current I is given by \[B = \frac{\mu_0 NI}{2\pi R} = \mu_0 n I\]where $ n = \frac{N}{2\pi R} $ is the number of turns per unit length.

16. The magnetic field inside a hollow pipe (or tube ) of current is zero. 

17. Magnetic field B at the centre due to a current flowing in a circular arc shaped conductor is  $ \frac{θ}{2π} $ times the magnetic field due to a circular loop, where θ is the angle subtended by the arc at the centre. Thus, \[B = \frac{\theta}{2\pi} \frac{\mu_0 I}{2R} = \frac{l}{2\pi R}\frac{\mu_0 I}{2R} = \frac{\mu_0 Il}{4\pi R^2}\] Where l is the length of conducting arc.

18. The force $\vec{F_B}$  acting on a electric charge q moving in a magnetic field $\vec{B}$ with the velocity $\vec{v}$ is called the magnetic Lorentz force and is given by \[\vec{F} = q (\vec{v}*\vec{B} )\]or \[F = qvBsin\theta\]where θ is the angle between the direction of v and the magnetic field B as given by right hand rule. Lorentz force is non-conservative force.

19. Magnitude of Lorentz magnetic force is determined by the component of velocity of direction perpendicular to that of magnetic field B. 

(i)if angle θ between B and v is 0⁰ or 180⁰ i.e., charged particle in moving parallel or antiparallel to the magnetic field, force F_B = 0.

(ii) if motion of charged particle in a direction perpendicular to that of magnetic field, force acting, on its maximum having a value F_B = q v B.

20. The charged particle entering a uniform magnetic field B, in a direction perpendicular to that of B, with a velocity v moves along a circular path of radius ‘r’ in a plane at right angle to B given by \[r = \frac{mv}{qB} = \frac{p}{qB} = \frac{\sqrt{2mK}}{qB}\]Where p = momentum of charged particle and K its kinetic energy. 

21. The magnetic force behave as the centripetal force and does not do any work. Thus, it does not change the kenetic energy of moving charge. However, due to change in direction of motion the velocity and momenta of charged particle change. The time period is complete one revolution in the circular trajectory is \[T = \frac{2\pi m}{qB}\]and the frequency of revolution is \[\nu = \frac{1}{T}=\frac{qB}{2\pi m}\]The frequency is independent of the charged particle’s speed as well as radius.

22. A charged particle entering a uniform magnetic field B in a direction making an angle θ from B, describes a helical path of radius r given by \[r = \frac{mvsin\theta }{qB}\]The pitch p of a helical path (I.e., linear distance covered in one revolution along the direction of B ) is given by \[p = \frac{2\pi mvcos\theta }{qB}\]

23. When a charged particle q moves along the direction of an electric field $ \vec{E} $, its motion is accelerated or retarded depending on the sign of charge q. However, the path of charged particle remain a straight line. When a charged particle is allowed to entre an electric field in a direction perpendicular to that of electric field, path of the particle is a parabolic path.

24. A charge q moving simultaneously in an electric and a magnetic field experience a force called total Lorentz force F, given by \[\vec{F} =\vec{F_B} + \vec{F_E} = q(\vec{v}*\vec{B}) + q\vec{E} = q[(\vec{v}*\vec{B}) + \vec{E}]\]If electric field $ \vec{E} $ and magnetic field $ \vec{B} $ are mutually perpendicular to each other as well as perpendicular to $ \vec{v} $ then $\vec{F_B}$ and $\vec{F_E}$ are along same straight line. If we adjust values of E and B such that magnitudes of F_B and F_E are equal and opposite, then net force on charge is zero and the charged particle goes undeviated. It happens, when \[qvB = qE \implies v = \frac{E}{B}\] 

25. A cyclotron uses both electric and magnetic fields, mutually crossed one, to accelerate charged particles or ions to high energies. Under a magnetic field the charged particle describes circular paths but after every half revolution it is suitably accelerated by the oscillating electric field operating at cyclotron frequency, whose value is given by \[\nu_c = \frac{qB}{2\pi m}\]The maximum K.E. of ion beam obtained from a cyclotron is \[K_{max} = \frac{1}{2}mv_{max}^2 = \frac{q^2B^2R^2}{2m}\]where R is the radius of the Dees of the cyclotron.

26. A current element $ I \vec{dl} $ when placed a uniform magnetic field $ \vec{B} $ experience a mechanical force \[\vec{F} = I(\vec{dl} * \vec{B} )\]For a straight linear conductor \[\vec{F} = I(\vec{l}*\vec{B})\]and \[F = IlBsin\theta\]The force $ \vec{F} $ is in a plane perpendicular to both $ \vec{dl} $ (or $\vec{I} $ ) and $\vec{ B} $ in the direction of their cross product. Direction of mechanical force acting on a current carrying plane conductor can be noted with the help of Fleming’s left hand rule. According to it stretch out the central finger, forefinger and thumb of your left hand to be mutually perpendicular to each other. If centre finger points in the direction of current and the forefinger in the direction of magnetic field then the thumb will point in the direction of the force.

27. Two parallel, straight, long, current carrying conductors attract each other if the current flowing in them in the same direction but repel each other if the currents are in mutually opposite directions. The force of interaction per unit length is \[\frac{F}{l} = \frac{\mu_0}{4\pi} \frac{2I_1I_2}{d}\]where d is the separation between the wires. 

28. A current flowing in a closed loop (either circular or of any other shape) produces a magnetic field pattern like that produced by a magnetic dipole. The magnetic moment $\vec{m} $ of a current loop is given by \[\vec{m} = I\vec{A}\]and for a coil of N turns \[\vec{m} = N I \vec{A}\]where A = area vector of the loop and NI = number of ampere turns in the coil. The direction of magnetic moment is given by right hand rule. SI unit of magnetic dipole moment in ampere meter² (A m²).

29. A current loop produced a magnetic field and behaves like a magnetic dipole at large distances. Moreover, a current loop is subject to torque like a magnetic needle, hence we conclude that ordinarily all magnetism is due to circulating current. When a coil carrying current of magnetic dipole moment \[\vec{m} = N I \vec{A}\]is placed in an orientation θ with a uniform magnetic field B , it experiences a torque given by \[\vec{\tau} = \vec{m}*{\vec{B}} , \tau = mBsin\theta = NIABsin\theta\]For a radical magnetic field, when magnetic field lines are perpendicular to the arms of a rectangular coil in every orientation of the coil, θ = 90⁰ and, hence \[\tau = mB = NAIB\]

30. A moving coil galvanometer is an extremely sensitive device and measure electric current flowing in a current when placed in a radial magnetic field. Current can be measured by using the formula \[I = \frac{k}{NAB}\phi\]where k = torsional constant of the suspension fibre (or spring) of galvanometer and Ф = angular deflection of the coil on passing current through it. The term $ \frac{Ф}{I} $ i.e., deflection per unit current is known as the current sensitivity of a galvanometer. Current sensitivity \[\frac{\phi}{I} = \frac{N A B}{k}\]Thus, to enhance the current sensitivity we use a strong magnetic field B and suspension fibre (or spring ) of small value of torsional constant. However, increase in N and A is not possible beyond a limit due to practical problems.

31. A galvanometer can be converted into an ammeter of an appropriate range by connecting a suitable, small shunt resistance in parallel to the galvanometer. If a galvanometer having resistance R_G  and giving full scale deflection for a current I_g is to be converted into an ammeter of range I ampere, then shunt resiistance $r_s$ used in parallel is given by \[r_s = \frac{R_G I}{I-I_g}\]The net resistance of ammeter is \[\frac{R_G* r_s}{R_G + r_s }\]which is extremely small. Ammeter is always joined in series of the electrical curcuit in which is to be measured.

32. A galvanometer may be converted into a voltmeter of given range V by joining a suitable high resistance R in series of galvanometer, such that \[R = \frac{V}{I_g} - R_G\]The net resistance of a voltmeter is (R + R _G), which is quite high. A voltmeter is always connected in parallel of an electrical circuit across the points, potential difference between which is to be measured.

33. A revolving electron in an orbit around a nucleus constitutes a current and there will be an orbital magnetic moment $\mu_l$ having a magnitude given by \[\mu_l = \frac{n e h}{4\pi m_e }\]where n = 1,2, 3,…etc. In vector notations, we have \[\vec{\mu_l} = -\frac{ e }{2 me }\vec{l}\]where $\vec{l} $ is the orbital angular momentum of electron $( l = m_evr) $. The negative sign indicates that the angular momentum of the electron is opposite in direction to orbital magnetic moment.

34. The minimum value of orbital magnetic moment of a revolving electron is given by \[\mu_{min} = \frac{e h }{4\pi m_e }\] (when n = 1) and has a value 9.27 x 10^-24 A m². This term \[\mu_{min} = \frac{e h}{4\pi m_e } = 9.27 *10^{-24} A m^2\]is known as Bohr magneton. Beside the orbital moment, the electron has a intrinsic magnetic moment. It is called the spin magnetic moment.  

    

Note:  "*" indicate vector product of two physical quantities.

 

   

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