Question

A dielectric sphere carries a uniform polarization \(P = 26\ \mu\text{C}.\text{cm}^{-2}\). The magnitude of the electric field at the center of the sphere is \(E \times 10^{9}\ \text{N}.\text{C}^{-1}\). The value of \(E\) (rounded off to one decimal place) is . \((\epsilon_0 = 8.85 \times 10^{-12}\ \text{C}^{2}.\text{N}^{-1}.\text{m}^{-2})\)

Hint:

Hint:
Inside a uniformly polarized sphere, \(E = \dfrac{P}{3\epsilon_0}\); convert \(P\) to C/m\(^2\) before plugging in.

Answer 1

Correct Answer: 9.8

Step 1: Understanding the Concept:
A uniformly polarized dielectric sphere produces a uniform electric field inside itself, pointing opposite to the polarization vector \(\vec{P}\). This is a standard result from the theory of bound charges: the surface bound charge on a uniformly polarized sphere creates a field that is uniform everywhere inside, including at the center.

Step 2: Key Formula or Approach:
The field inside a uniformly polarized sphere is:
\[ \vec{E}_{in} = -\dfrac{\vec{P}}{3\epsilon_0} \]
so its magnitude is \(E = \dfrac{P}{3\epsilon_0}\). We need \(P\) in SI units (C/m\(^2\)) and then divide by \(3\epsilon_0\).

Step 3: Detailed Explanation:
Convert \(P\) from \(\mu\text{C/cm}^2\) to SI units. Since \(1\ \text{cm}^2 = 10^{-4}\ \text{m}^2\):
\[ P = 26\ \mu\text{C/cm}^2 = 26\times10^{-6}\ \text{C} \times \dfrac{1}{10^{-4}\ \text{m}^2} = 0.26\ \text{C/m}^2 \]
Now compute \(E\):
\[ E = \dfrac{P}{3\epsilon_0} = \dfrac{0.26}{3\times8.85\times10^{-12}} = \dfrac{0.26}{2.655\times10^{-11}} \approx 9.79\times10^{9}\ \text{N/C} \]

Step 4: Final Answer:
Comparing with \(E\times10^9\ \text{N/C}\), we get \(E \approx 9.79\), which rounds to one decimal place as \(9.8\). \[ \boxed{E = 9.8} \]