Question

The capacitance of a capacitor becomes \(\frac{7}{6}\) times the original value if dielectric slab of thickness \(t = \frac{2d}{3}\) is introduced between the plates, where 'd' is the distance of separation between the plates. The dielectric constant of the slab is

• A. \(\frac{9}{11}\)
• B. \(\frac{14}{11}\)
• C. \(\frac{8}{11}\)
• D. \(\frac{12}{11}\)

Hint:

With a partial dielectric slab, the capacitance is epsilon naught A divided by (d - t + t/K). Set it equal to 7/6 of the original.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A dielectric slab of thickness \(t\) and constant \(K\) in a gap \(d\) makes the effective gap \(d - t + \dfrac{t}{K}\).

Step 2: Key Formula or Approach:
\[ C = \frac{\varepsilon_0A}{d - t + t/K}, \qquad C_0 = \frac{\varepsilon_0A}{d} \]

Step 3: Use the condition.
\[ \frac{C}{C_0} = \frac{d}{d - t + t/K} = \frac{7}{6} \Rightarrow d - t + \frac{t}{K} = \frac{6d}{7} \]
Put \(t = \dfrac{2d}{3}\): \(d - \dfrac{2d}{3} + \dfrac{2d}{3K} = \dfrac{6d}{7}\), so
\[ \frac{2}{3K} = \frac{6}{7} - \frac{1}{3} = \frac{18 - 7}{21} = \frac{11}{21} \]

Step 4: Solve for K.
\[ K = \frac{2}{3}\times\frac{21}{11} = \frac{14}{11} \]

Step 5: Check.
Strictly a dielectric constant should be at least 1, and \(14/11 \approx 1.27\) satisfies that. The other options \(9/11\), \(8/11\), \(12/11\) do not satisfy the equation.

Final Answer:
The dielectric constant is \(\dfrac{14}{11}\), option (B). \[ \boxed{K = \frac{14}{11}} \]