A parallel plate capacitor with plate area ' $A$ ' and distance of separation ' $d$ ' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as:
$ε (x) = ε_{0} + k x$ , for $(0 < x \leq \frac{d}{2})$
$ε (x) = ε_{0} + k (d - x)$ , for $(\frac{d}{2} \leq x \leq d)$

Question

A parallel plate capacitor with plate area ' $A$ ' and distance of separation ' $d$ ' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as:

$ε (x) = ε_{0} + k x$ , for $(0 < x \leq \frac{d}{2})$

$ε (x) = ε_{0} + k (d - x)$ , for $(\frac{d}{2} \leq x \leq d)$

Aicharya · Accepted Answer

Taking an element of width dx at a distance $x (x < d / 2)$ from left plate

$d c = \frac{(ε_{0} + k x) A}{d x}$

Capacitance of half of the capacitor

$\frac{1}{C} = \int_{0}^{d / 2} \frac{1}{d c} = \frac{1}{A} \int_{0}^{d / 2} \frac{d x}{ε_{0} + k x}$

$\frac{1}{C} = \frac{1}{k A} l n (\frac{ε_{0} + k d / 2}{ε_{0}})$

Capacitance of second half will be same

$C_{e q} = \frac{C}{2} = \frac{k A}{2 l n (\frac{2 ε_{0} + k d}{2 ε_{0}})}$

A parallel plate capacitor with plate area ' A ' and distance of separation ' d ' is filled…

Solution

A parallel plate capacitor with plate area &#39; A &#39; and distance of separation &#39; d &#39; is filled…

Solution

A parallel plate capacitor with plate area ' A ' and distance of separation ' d ' is filled…