Question

Parallel plate capacitor... separation 5 mm... mica sheet 2 mm... draws 25\% more charge. Dielectric constant is ___.

• A. 2.0
• B. 1.0
• C. 1.5
• D. 2.5

Hint:

$C_{new} = \frac{\epsilon_0 A}{d-t(1-1/K)}$.

Answer 1

The Correct Option is A

A parallel plate capacitor initially has no dielectric material between its plates. When a mica sheet of thickness 2 mm is inserted (the total separation between the plates is 5 mm), it is stated that the capacitor draws 25% more charge. We need to find the dielectric constant of mica.

The capacitance of a capacitor without a dielectric is given by:

\(C_0 = \frac{\varepsilon_0 A}{d}\)

where: 

• \(\varepsilon_0\) is the permittivity of free space,
• \(A\) is the area of the plates, and
• \(d\) is the separation between the plates.

 

With the dielectric (mica) inserted, the capacitor system now consists of two dielectric mediums: air and mica. The mica fills 2 mm of the space, while the air fills the remaining 3 mm. The effective capacitance \(C\) is given by:

\(C = \frac{\varepsilon_0 A}{d_{air}} + \frac{\varepsilon_ A}{d_{mica}}\)

where

• \(d_{air} = 3 \, \text{mm}\),
• \(d_{mica} = 2 \, \text{mm}\), and
• \(\varepsilon\) is the permittivity of the mica, which can be expressed as \(\varepsilon = K \cdot \varepsilon_0\), where \(K\) is the dielectric constant of mica.

The new capacitance becomes:

\(C = \varepsilon_0 A \left(\frac{1}{d_{air}} + \frac{K}{d_{mica}}\right)\).

We know the charge increases by 25%, meaning

\(C = 1.25 C_0\).

Substituting the values, we have:

\(\varepsilon_0 A \left(\frac{1}{3 \times 10^{-3}} + \frac{K}{2 \times 10^{-3}}\right) = 1.25 \times \frac{\varepsilon_0 A}{5 \times 10^{-3}}\).

Simplifying, we find:

\(\left(\frac{1}{3 \times 10^{-3}} + \frac{K}{2 \times 10^{-3}}\right) = \frac{1.25}{5 \times 10^{-3}}\).

Solving for \(K\),

\(\frac{1}{3} + \frac{K}{2} = \frac{1.25}{5} = 0.25\).

\(K = \frac{5}{2} - \frac{2}{3} = \frac{10 - 2}{3} = 2\).

Therefore, the dielectric constant of mica is \(2.0\).

Hence, the correct answer is 2.0.