Question

The energy stored in the electric field produced by a metal sphere is $4.5\, J$. If the sphere contains $ 4 \, \mu C$ charge, its radius will be : [Take : $\frac{1}{4 \pi \epsilon_o } = 9 \times 10^{9} \, N - m^2 / C^2$ ]

• A. 20 mm
• B. 32 mm
• C. 28 mm
• D. 16 mm

Answer 1

The Correct Option is D

To find the radius of a metal sphere given that the energy stored in its electric field is \(4.5\, J\) and it contains a charge of \(4 \, \mu C\), we can use the formula for the electric field energy stored in a sphere: 

\(E = \frac{1}{8\pi\epsilon_0} \cdot \frac{Q^2}{R}\)

From the question, we're given:

• Energy, \(E = 4.5\, J\)
• Charge, \(Q = 4 \, \mu C = 4 \times 10^{-6}\, C\)
• \(\frac{1}{4 \pi \epsilon_0} = 9 \times 10^{9} \, N \cdot m^2 / C^2\)

 

We can also express \(E\) as:

\(E = \frac{1}{2} \cdot \frac{1}{4\pi\epsilon_0} \cdot \frac{Q^2}{R}\)

Rearranging the above expression to solve for \(R\), we have:

\(R = \frac{1}{2} \cdot \frac{1}{4\pi\epsilon_0} \cdot \frac{Q^2}{E}\)

Substitute the given values:

\(R = \frac{1}{2} \cdot 9 \times 10^{9} \cdot \frac{(4 \times 10^{-6})^2}{4.5}\)

Calculate the value inside the square:

\((4 \times 10^{-6})^2 = 16 \times 10^{-12}\)

Substitute back into the equation for \(R\):

\(R = \frac{1}{2} \cdot 9 \times 10^{9} \cdot \frac{16 \times 10^{-12}}{4.5}\)

\(R = \frac{1}{2} \cdot 9 \times \frac{16}{4.5} \times 10^{-3} \, m\)

\(R \approx \frac{1}{2} \cdot 32 \times 10^{-3} \, m\)

\(R \approx 16 \times 10^{-3} \, m = 16 \, mm\)

The radius of the sphere is \(16\, mm\).

Therefore, the correct answer is: 16 mm.