# Numerical Problems : Capacitor

##### LEVEL – II

Q: 1. If the area of parallel plates shown in the figure is ‘ A ‘ and they are placed at distance ‘ d ‘ apart form each other, then find the equivalent capacitance between A and B. The two outer plates are connected with a conducting wire. Ans: $\displaystyle \frac{\epsilon_0 A}{d} (\frac{k_1 k_2 + k_1 + k_2}{k_1 + k_2} )$

Q: 2. A capacitor of capacitance 0.1 μ F is charged until the difference in potential between its plates is 25 V. Then the charge is shared with a second capacitor which has air as dielectric. The potential difference falls to 15 V. If the experiment is repeated with dielectric introduced between the plates of the second capacitor, the potential difference is 8 V. What is the dielectric constant of the material introduced ?

Ans: 3.2

Q: 3. Determine the potential difference φA – φB between points A and B of the circuit shown in figure. Under what condition is it equal to zero? Ans: $\displaystyle \phi_A – \phi_B = E \frac{C_2 C_3 – C_1 C_4}{(C_1 + C_2)(C_3 + C_4)}$ ;

When C1/C2 = C3/C4

Q: 4. Two metal plates form a parallel plate capacitor. The distance between the plates is given as ‘ d ‘ . A metal plate of thickness (d/2), and two dielectric slabs of thickness (d/4) is introduced between the plates as shown in the figure. If the metal plate is removed find the work done in slowly removing it. (The plates of capacitor is connected to a battery having potential difference ‘ v ‘ ) Ans: $\displaystyle \frac{4 \epsilon_0 A V^2 k_1^2 k_2^2}{d(k_1 + k_2)(k_1 + k_2 + 2 k_1 k_2)}$

Q:5. Find the equivalent capacitance between A and B in the circuit shown below. If the ends ‘ A ‘ and ‘ B ‘ are connected across a 12 V cell, find the electrostatic potential energy of the system. (the capacitance of each capacitor is 100 μF) Ans: 125 μF, 9000 μ J

Q:6. Each capacitor has a capacitance of 5 μF. Find the charge that will flow through MN when the switch S is closed. Ans : 333.3 μC

Q:7. In the figure shown , determine the potential differences on the plates of capacitors C1 = 3μF , C2 = 7μF , if value of E1 = 12kV , E2 = 13kV Ans: 700 V, 300 V

Q:8. Find the equivalent capacitance between A and B , if the plates have equal area A and the separation between the plates is d Ans: $\displaystyle \frac{2 \epsilon_0 A}{d}$

Q:9. A uniform electric field E exists between the plates of a capacitor. The plate length is l and the separation of the plates is d.

(a) An electron and a proton start from the negative plate and positive plate respectively and go to the opposite plates. Which of them wins this race?

(b) An electron and a proton are projected parallel to the plates from the midpoint of the separation of plates at one end of the plates. Which of the two will have greater deviation when they start with the

(i) same initial velocity

(ii) same initial kinetic energy, and

(iii) same initial momentum ?

Ans: (a) Electron (b) (i) Electron (ii) Both equal deviation (iii) Proton

Q:10. Figure shows a parallel plate capacitor having square plates of edge a and plate separation d. The gap between the plate is filled with a dielectric of dielectric constant k which varies from the left plate to the right plate as k = ko + αx , where ko and α are positive constants and x is the distance from the left end. Calculate the capacitance Ans: $\displaystyle \frac{\epsilon_0 a^2 \alpha}{ln(1+\frac{\alpha d}{k_0})}$

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