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

To solve this problem, we need to understand the behavior of a parallel plate capacitor when a dielectric is added. We must assess how the dielectric constant affects the charge stored in the capacitor.

First, let's recall the formula for the capacitance \(C\) of a parallel plate capacitor:

\(C = \frac{\varepsilon_0 \cdot A}{d}\) 

where:

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

When a dielectric of constant \(k\) and thickness \(t\) is inserted between the plates, the effective capacitance \(C_{eff}\) becomes:

\(C_{eff} = \frac{\varepsilon_0 \cdot A}{d - t + \frac{t}{k}}\)

The problem states that the capacitor draws 25% more charge with the dielectric inserted. This means the new capacitance is 1.25 times the original capacitance, because charge \((Q)\) is directly proportional to capacitance:

\(Q = C \cdot V\). Since voltage \((V)\) remains constant, the increase in charge is due to the increase in \(C\).

Hence, we have:

\(\frac{C_{eff}}{C} = 1.25\)

Substituting the expressions for \(C_{eff}\) and \(C\), we get:

\(\frac{\frac{\varepsilon_0 \cdot A}{d - t + \frac{t}{k}}}{\frac{\varepsilon_0 \cdot A}{d}} = 1.25\)

Canceling out common terms, we simplify to:

\(\frac{d}{d - t + \frac{t}{k}} = 1.25\)

Given:

• \(d = 5 \text{ mm} = 5 \times 10^{-3} \text{ m}\)
• \(t = 2 \text{ mm} = 2 \times 10^{-3} \text{ m}\)

Substituting these values into the equation gives:

\(\frac{5 \times 10^{-3}}{5 \times 10^{-3} - 2 \times 10^{-3} + \frac{2 \times 10^{-3}}{k}} = 1.25\)

Simplifying further:

\(5 \times 10^{-3} = 1.25 \cdot \left(3 \times 10^{-3} + \frac{2 \times 10^{-3}}{k}\right)\)

Dividing through by \(10^{-3}\):

\(5 = 1.25 \cdot \left(3 + \frac{2}{k}\right)\)

Simplifying gives:

\(5 = 3.75 + \frac{2.5}{k}\)

Solving for \(k\):

\(1.25 = \frac{2.5}{k}\)

\(k = \frac{2.5}{1.25} = 2.0\)

Thus, the dielectric constant is \(2.0\).

The correct answer is therefore 2.0.