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JEE Main20199 Jan 2019Morning ShiftPhysicsCapacitanceActual

A parallel plate capacitor is made of two square plates of side a , separated by a distance d d ≪ a . The lower triangular portion filled with a dielectric of dielectric constant K , as shown in the figure. Capacitance of this capacitor is:

Options

  1. AK ε 0 a 2 2 d K + 1
  2. BK ε 0 a 2 d K - 1 ln ⁡ K
  3. CK ε 0 a 2 d ln ⁡ K
  4. D1 2 K ε 0 a 2 d

Correct answer

B. K ε 0 a 2 d K - 1 ln ⁡ K

Step-by-step solution

Consider a small element at a distance x. It is a two capacitors in series. d c = d c 1 × d c 2 d c 1 + d c 2 d c = ε 0 . a d x x t a n θ k + d - x t a n θ 1 ∴ d c = ε 0 K . a d x - x tan ⁡ θ K - 1 + d . k ∴ C = ∫ 0 a ε 0 K . a d x - x tan ⁡ θ K - 1 + d . k = ε 0 k . a tan ⁡ θ ( K - 1 ) l n   - a tan ⁡ θ K - 1   + d . k d . K = ε 0 K a 2 d K - 1 l n   K

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