Question

A charge of −9C is placed at the center of a concentric spherical shell made of a linear dielectric material (relative permittivity 9) and having inner and outer radii of 0.1m and 0.2m, respectively. The total charge induced on its inner surface is ______C. (Rounded off to two decimal place)

Answer 1

Correct Answer: 7.9

Given Data 

• Charge at the center: \( Q = -9C \)
• Relative permittivity of the dielectric: \( \varepsilon_r = 9 \)
• Inner radius of the shell: \( r_1 = 0.1m \)
• Outer radius of the shell: \( r_2 = 0.2m \)

Step 1: Formula for Induced Charge

The total charge induced on the inner surface of the dielectric shell is given by:

\[ Q_{\text{induced}} = -Q \left( 1 - \frac{1}{\varepsilon_r} \right) \]

Step 2: Substituting the Given Values

Substituting \( Q = -9C \) and \( \varepsilon_r = 9 \):

\[ Q_{\text{induced}} = -(-9) \left( 1 - \frac{1}{9} \right) \]

Step 3: Simplifying the Expression

\[ Q_{\text{induced}} = 9 \left( 1 - \frac{1}{9} \right) \]

\[ Q_{\text{induced}} = 9 \left( \frac{9 - 1}{9} \right) \]

\[ Q_{\text{induced}} = 9 \times \frac{8}{9} \]

\[ Q_{\text{induced}} = 8C \]

Step 4: Rounding Off

Since the calculation rounds off to two decimal places, we write:

\[ Q_{\text{induced}} \approx 8.00C \]

Final Answer

Thus, the total charge induced on the inner surface of the dielectric shell is \( 8.00C \).