JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Capacitance

Four identical rectangular plates with length,

ℓ = 2 cm

and breadth,

b = \frac{3}{2} cm

are arranged as shown in figure. The equivalent capacitance between

A

and

C

\frac{X ε _{0}}{d}

. The value of

x

is ⋯------. (Round off to the nearest integer)

Answer: 2

Solution

Each neighbouring pair of plates a distance

d

apart forms a capacitor of capacitance

C_{0} = \frac{ε _{0} ℓ b}{d}

, and here the plate area is

ℓ b = 2 \times \frac{3}{2} = 3 c m^{2}

.
The wire joining

B

and

D

makes them a single node. Between the terminals

A

and

C

the pair

A

B

gives

C_{0}

in series with the parallel combination of the pairs

B

C

and

C

D

, each also

C_{0}

C_{A C} = \frac{C _{0} ( 2 C _{0} )}{C _{0} + 2 C _{0}} = \frac{2 C _{0}}{3}

C_{A C} = \frac{2}{3} \cdot \frac{3 ε _{0}}{d} = \frac{2 ε _{0}}{d}

Comparing with

\frac{x ε _{0}}{d}

gives

x = 2

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Integer answer