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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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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