To find the electric potential inside the conducting spherical shell, we need to use the relationship between the electric field and electric potential.

Step 1: Electric Field inside the conducting shell

Given that the electric field E inside the spherical shell (for $r < R$) is zero (since the shell is a conductor and the charges reside on the outer surface), we have:

$E(r) = 0 \quad \text{for} \quad r < R$

Step 2: Electric Potential and Electric Field Relationship

The electric potential $V(r)$ is related to the electric field $E(r)$ by the following equation:

$V(r) = - \int_{\infty}^{r} E(r') \, dr'$

Step 3: Electric Field Inside the Sphere

Since $E(r) = 0$ for $r < R$, the potential inside the spherical shell must be constant. The electric potential inside the shell will be equal to the potential at the surface $r = R$, because there is no electric field to change the potential inside the shell.

Step 4: Calculate Potential at the Surface

The potential at the surface of the sphere, $V(R)$, can be computed by considering the electric field outside the shell (for $r > R$):

$E(r) = \frac{1}{4\pi \epsilon_0} \frac{q}{r^2} \quad \text{for} \quad r > R$

Now, we calculate the potential at the surface $V(R)$:

$V(R) = - \int_{\infty}^{R} E(r') \, dr'$

Substitute the expression for the electric field:

$V(R) = - \int_{\infty}^{R} \frac{1}{4\pi \epsilon_0} \frac{q}{r'^2} \, dr'$

This integral evaluates to:

$V(R) = - \frac{q}{4\pi \epsilon_0} \left[ \frac{1}{R} - \frac{1}{\infty} \right] = \frac{q}{4\pi \epsilon_0 R}$

Step 5: Electric Potential Inside the Shell

Since the electric field is zero inside the shell, the potential inside is the same as at the surface. Therefore, the electric potential inside the shell (for $r < R$) is:

$V(r) = \frac{q}{4\pi \epsilon_0 R}$

Final Answer:

$V(r) = \frac{q}{4\pi \epsilon_0 R} \quad \text{for} \quad r < R$

Where:

• $q = 31 \, \mu C = 31 \times 10^{-6} \, C$
• $R = 5 \, \text{m}$
• $\epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}$

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Would you like any further details or have additional questions on this problem?

Here are 5 related questions:

1. How would the electric potential change if the shell were not conducting?
2. How would the result change if we had a uniformly charged solid sphere instead of a shell?
3. What would be the potential outside the spherical shell (for $r > R$)?
4. How would the electric field and potential behave if the shell had a non-zero surface charge distribution?
5. What is the electric field inside a conducting sphere for $r < R$?

Tip: Always remember that for a conductor in electrostatic equilibrium, the electric field inside is zero, and the potential inside is constant and equal to the potential at the surface.