JEE Advanced 2022 Paper 1 Q09 Physics Electrostatics & Circuits Capacitors with Dielectric Hard

JEE Advanced 2022 Paper 1 · Q09 · Capacitors with Dielectric

A medium having dielectric constant K > 1 fills the space between the plates of a parallel plate capacitor. The plates have large area, and the distance between them is d. The capacitor is connected to a battery of voltage V, as shown in Figure (a). Now, both the plates are moved by a distance of d/2 from their original positions, as shown in Figure (b). In the process of going from the configuration depicted in Figure (a) to that in Figure (b), which of the following statement(s) is(are) correct?

[Figure (a): Parallel plates separated by d with dielectric (K) filling the gap, connected to battery V. Figure (b): plates moved apart by d/2 each (total separation now 2d) with the same slab of dielectric of thickness d in the middle and air gaps of d/2 on either side; same battery V connected.]

  1. A. The electric field inside the dielectric material is reduced by a factor of 2K.
  2. B. The capacitance is decreased by a factor of 1/(K+1).
  3. C. The voltage between the capacitor plates is increased by a factor of (K+1).
  4. D. The work done in the process DOES NOT depend on the presence of the dielectric material.
Reveal answer + step-by-step solution

Correct answer:B

Solution

Battery stays connected, so V across the plates remains V. In (a): C_a = Kε₀A/d, field in dielectric E_a = V/d. In (b): the gap is now 2d but only a slab of thickness d is dielectric; the two air gaps total d. Equivalent capacitance = 1/(d/(Kε₀A) + d/(ε₀A)) = ε₀A/d · K/(K+1). So C_b = (K/(K+1))·ε₀A/d = C_a/(K+1)... wait, C_a = Kε₀A/d, C_b = Kε₀A/[d(K+1)] ⇒ C_b/C_a = 1/(K+1). So capacitance is decreased by a factor of 1/(K+1) ⇒ (B) correct. (A) Field in dielectric in (b): Q_b = C_b V = KAε₀V/[d(K+1)]; field in dielectric E_b = Q_b/(Kε₀A) = V/[d(K+1)]. Ratio E_b/E_a = 1/(K+1), NOT 1/(2K). (A) incorrect. (C) Voltage stays at V (battery), so it does not change by (K+1); incorrect. (D) Work done by battery W = ΔU + W_ext; since charge changes, work done depends on K, so (D) is incorrect. Note: FIITJEE answer key lists (B) only.

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