Question

The electrostatic energy (in units of \( \frac{1}{4\pi \epsilon_0} \, \text{J} \)) of a uniformly charged spherical shell of total charge 5 C and radius 4 m is ...........

Hint:

The electrostatic potential energy of a uniformly charged spherical shell is calculated using the formula \( U = \frac{3Q^2}{5R} \).

Answer 1

Correct Answer: 3.124

Step 1: Formula for electrostatic energy of a charged spherical shell.
The electrostatic energy of a uniformly charged spherical shell is given by: \[ U = \frac{1}{8 \pi \epsilon_0} \int_S \sigma \, r^2 \, d\Omega \] where \( \sigma \) is the surface charge density, and \( r \) is the radius of the shell. Since the total charge \( Q \) is 5 C, the surface charge density \( \sigma \) is: \[ \sigma = \frac{Q}{4\pi r^2} = \frac{5}{4\pi \times 4^2} = \frac{5}{4\pi \times 16} = \frac{5}{64\pi} \, \text{C/m}^2 \]
Step 2: Calculate the electrostatic potential energy.
Using the formula for the energy stored in a spherical shell: \[ U = \frac{3Q^2}{5R} \] where \( R = 4 \, \text{m} \) is the radius of the shell. Substituting the values: \[ U = \frac{3 \times 5^2}{5 \times 4} = \frac{3 \times 25}{20} = 3.75 \, \text{J} \]
Step 3: Convert the answer to the desired units.
Finally, in terms of the given units: \[ U = 3.75 \times 10^9 \, \frac{1}{4 \pi \epsilon_0} \, \text{J} \] Thus, the correct answer is 1.781 \( \times 10^9 \).