If a dielectric plate of thickness t is placed between the plates of a parallel plate capacitor of plate distance d, the capacitance becomes half of the original value. The dielectric constant of the plate will be:

Question

If a dielectric plate of thickness t is placed between the plates of a parallel plate capacitor of plate distance d, the capacitance becomes half of the original value. The dielectric constant of the plate will be: (A)  (2t/2d+t)  (B)  (2t/2d-t)  (C)  (t/d+t)  (D)  (t/d-t)

Tardigrade Admin · Accepted Answer

The capacitor of the capacitance without placing the dielectric plate is given by

C = \frac{ε _{0} A}{d}

?(1)
When the dielectric plate of dielectric constant K and thickness t is inserted between the parallel plates of a capacitor, then it becomes the series combination of two capacitors as shown in figure, so the equivalent capacitance

C_{e q}

is

C_{e q} = \frac{C}{2} = \frac{C _{1} C _{2}}{C _{1} + C _{2}}

Here,

C_{1} = \frac{ε _{0} A}{d - t}

and

C_{2} = \frac{K ε _{0} A}{t}

or

\frac{C}{2} = \frac{( \frac{ε _{0} A}{d - t} ) \times ( \frac{K ε _{0} A}{t} )}{ε _{0} A { \frac{1}{d - t} + \frac{K}{t} }} = \frac{K ε _{0} A}{t + ( d - t ) K}

or

\frac{C}{2} = \frac{K ε _{0} A}{t + ( d - t ) K}

As

C = \frac{ε _{0} A}{d}

or

ε_{0} A = C d

or

\frac{C}{2} = \frac{K C d}{t + ( d - t ) K}

or

t + (d - t) K = 2 K d

or

K {- (d - t) + 2 d} = t

or

K = \frac{t}{d + t}

Q. If a dielectric plate of thickness t is placed between the plates of a parallel plate capacitor of plate distance d, the capacitance becomes half of the original value. The dielectric constant of the plate will be:

$\frac{2 t}{2 d + t}$

$\frac{2 t}{2 d - t}$

$\frac{t}{d + t}$

$\frac{t}{d - t}$

Q. If a dielectric plate of thickness t is placed between the plates of a parallel plate capacitor of plate distance d, the capacitance becomes half of the original value. The dielectric constant of the plate will be:

2d+t2t​

2d−t2t​

d+tt​

d−tt​

$\frac{2 t}{2 d + t}$

$\frac{2 t}{2 d - t}$

$\frac{t}{d + t}$

$\frac{t}{d - t}$