Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

JEE: Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
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Q.13. A circular ring of radius R with uniform positive charge density λ per unit length is located in the y-z plane with its centre at the origin O. A particle of mass m and positive charge q is projected from the point P (R√ 3,0,0) on the positive x-axis directly towards O, with an initial speed v.

Find the smallest (non-zero) value of the speed v such that the particle does not return to P.       (1993-4 Marks)

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Potential energy can be found at the initial point A and final point O. The difference in potential energy has to be provided by the K.E. of the charge at A.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Potential difference between points O and P = V

∴ V = VO-VP

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The kinetic energy of the charged particle is converted into its potential energy at O. 

∴ Potential energy of charge (q) = qV 

Kinetic energy of charged particle = 1/2 mv2

For minimum speed of particle so that it does not return to P,

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.14. Two square metal plates of side 1 m are kept 0.01 m apart like a parallel plate capacitor in air in such a way that one of their edges is perpendicular to an oil surface in a tank filled with an  insulating oil. The plates are connected to a battery of emf 500 V. The plates are then lowered vertically into  the oil at a speed of 0.001 ms–1. Calculate the current drawn from the battery during the process. (Dielectric constant of oil = 11, ε0 = 8.85 × 10–12C2N–1m–1)       (1994 - 6 Marks)

Ans. 4.425 × 10–9 A.

Solution. The adjacent figure is a case of parallel plate capacitor. The combined capacitance will be

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Differentiating the above quation w.r.t. time

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

where v = dx/dt

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From (iii) and (iv)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.15. The capacitance of a parallel plate capacitor with plate area A and separation d is C. The space between the plates is filled with two wedges of dielectric constants K1 and K2, respectively. Find the capacitance of the resulting capacitor.       (1996 - 2 Marks)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE 

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Case (i) When no dielectric :

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Case (ii) When dielectric is filled : A small dotted element of thickness dx is considered as shown in the figure.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The small capacitance of the dotted portion Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE capacitance of capacitor with dielectric K1

dC2 = capacitance of capacitor with dielectric K2.

Let ℓ, b the len gth an d br eadth of the capacitor plate.

Therefore ℓ × b=A.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

To find the capacitance of the whole capacitor, we integrate the above equation.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.16. Two capacitors A and B with capacities 3 μF and 2 μF are charged to a potential difference of 100 V and 180 V respectively. The plates of the capacitors are connected as shown in the figure with one wire from each capacitor free.

The upper plate of A is positive and that of B is negative. An uncharged 2 μF capacitor C with lead wires falls on the free ends to complete the circuit. Calculate         (1997 - 5 Marks)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(i) the final charge on the three capacitors. and 
(ii) the amount of electrostatic energy stored in the system before and after the completion of the circuit.

Ans. (i) 90 × 10–6 C, 210 × 10–6 C, 150 × 10–6 C  
(ii) 4.74 × 10–2 J, 1.8 × 10–2 J

Solution. (i) KEY CONCEPT : Use charge conservation to solve this problem.

INITIALLY :

Charge on capacitor A 

qA = 3  × 10–6 × 100 = 3 × 10–4C

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Charge on capacitor B 

qB = 2  × 10–6 × 180 = 3.6 × 10–4C

FINALLY :

Let the charge on capacitor A, C and B be q1, q2 and q3 respectively.

By charge conservation.

The sum of charge on +ve plate of capacitor A and C should be equal to qA

∴ q1 + q2 = 3 × 10–4C... (i)  

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Similarly the sum of charge on –ve plates of capacitor C and B will be equal to qB 

∴ – q2 – q3 = – 3.6 × 10–4

⇒ q2 + q3 = 3.6 × 10–4C ... (ii) 

Applying  Kirchoff's law in the closed loop, we get

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 2q1 – 3q2 + 3q3 = 0 ....(iii) 

On solving (i), (ii) and (iii), we get 

q1 = 90 × 10–6 C, q2 = 210 × 10–6 C, and q3 = 150 × 10–6 C, 

(ii) Amount of electrostatic energy in the system initially

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Amount of electrostatic energy stored finally

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE 

Q.17. A conducting sphere S1 of radius r is attached to an insulating handle. Another conducting sphere S2 of radius R is mounted on an insulating stand. S2 is initially uncharged.

S1 is given a charge Q, brought into contact with S2, and removed. S1 is recharged such that the charge on it is again Q; and it is again brought into contact with S2 and removed.

This procedure is repeated n times.             (1998 - 8 Marks) 

(a) Find the electrostatic energy of S2 after n such contacts with S1.
(b) What is the limiting value of this energy as n →∞ ?

Ans.Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.  Limiting value of energy as n →∞.

Let us calculate qn when n tends to ∞.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.18. A non-conducting disc of radius a and uniform positive surface charge density σ is placed on the ground, with its axis vertical. A particle of mass m and  positive charge q is dropped, along the axis of  the disc, from a height H with zero initial velocity. The particle has q/m = 4 ∈0g/σ    (1999 - 10 Marks) 

(a) Find the value of H if the particle just reaches the disc. 
(b) Sketch the potential energy of the particle as a function of its height and find its equilibrium position

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. (a) KEY CONCEPT :  The K.E. of the particle, when it reaches the disc is zero.

Given that a = radius of disc, σ = surface charge density, q/m = 4ε0g/σ 

Potential due to a charged disc at any axial point situated at a distance x from O is,

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

NOTE : According to law of conservation of energy, loss of gravitational potential energy = gain in electric potential energy

mgH = qΔV 

= q [V(0) – V (H)]

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From the given relation Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Putting this in equation (1), we get,

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE [∵H = O is not valid]

(b) Total potential energy of the particle at height H 

U (x) = mgx + qV (x)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From equation (2), graph between U(x) and x is as shown above.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.19. Four point charges +8mC, –1mC, –1mC, and +8mC are fixed at the points Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE respectively on the y-axis. A particle of mass 6 × 10–4 kg and charge +0.1 μC moves along the -x direction. Its speed at x = + ∞ is V0. Find the least value of V0 for which the particle will cross the origin. Find also the kinetic energy of the particle at the origin. Assume that space is gravity free.
Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE                      (2000 - 10  Marks)

Ans. 3 m/s,  3 × 10–4 J

Solution. Let the particle at some instant be at a point P distant x from the origin. As shown in the figure, there are two forces of repulsion acting due to two charges of + 8 mC. The net force is 2F cos α towards right.

Similarly there are two forces of attraction due to two charges of – 1 mC. The net force due to these force is 2F' cos β towards left.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The net force on charge 0.1 µC is zero when 2F cos α = 2F' cos β

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

This means that we need to move the charge from – ∞ to Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEEThereafter the attractive forces will make the charge move to origin. 

The electric potential of the four charges at Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE  is

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Kinetic energy is required to overcome the force of repulsion

from ∞ to Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The work done in this process is W = q (V) 

where V = p.d between ∞ and Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

K.E. at the origin 

Potential at origin

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

= 2.4 × 104

Again by energy conservation

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

∴K.E. = 0.1 × 10–6 [2.7 × 104 – 2.4 × 104]

= 0.1 × 10–6 × 0.3 × 104

= 3 × 10–4 J

Q.20. Charges +q and –q are located at the corners of a cube of side as show in the figure. Find the work done to separate the charges to infinite distance.          (2003 - 2 Marks)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.21. A charge +Q is fixed at the origin of the co-ordinate system while a small electric dipole of dipole moment Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEEpointing away from the charge along the x-axis is set free from a point far away from the origin.       (2003 - 4 Marks) 

(a) Calculate the K.E. of the dipole when it reaches to a point (d, 0). 

(b) Calculate the force on the charge +Q at this moment.

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. (a) Potential energy of the dipole-charge system 

Ui = 0 (since the charge is far away)

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(b) Electric field at origin due to dipole

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Thus, force on charge Q is given by

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.22. Two uniformly charged large plane sheets S1 and S2 having charge densities σ1 and σ21 > σ2) are placed at a distance d parallel to each other. A charge q0 is moved along a line of length a(a < d) at an angle 45º with the normal to S1. Calculate the work done by the electr ic field         (2004)

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Work done by electric field

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.23. A conducting liquid bubble of r adius a and th ickn ess t (t << a) is charged to potential V. If the bubble collapses to a droplet, find the potential on the droplet.        (2005 - 2  Marks)

Ans. Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. LIQUID BUBBLE : The potential of the liquid bubble is V.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

where q is the charge on the liquid bubble.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

LIQUID DROPLET

The volume of liquid droplet = Volume (of the liquid)   in liquid bubble.

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

or, r3 = a3 + t3 + 3a2t + 3at2 – a3 

or, r3 = 3a2t

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(Q t is very small as compared to a) 

or, r = [3a2t]1/3 ... (iii)

NOTE : By charge conservation we can conclude that charge on liquid bubble is equal to charge on liquid droplet Let charge on liquid droplet is q. 

∴ Potential on liquid droplet

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE [From (i) and (ii)]

Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Electrostatics- 2 | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
All you need of JEE at this link: JEE

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