Question

A hollow conducting sphere of radius \(R\) is centered about the origin of an \(x y z\) -coordinate system. A total charge \(Q\) is distributed uniformly over the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the sphere? a) zero b) \(2 k Q / R\) c) \(k Q / R\) d) \(k Q / 2 R\) e) \(k Q / 4 R\)

Short answer

Short Answer: The electric potential at the center of the sphere is given by \(V(0) = \frac{kQ}{R}\), where k is the electrostatic constant.

Step-by-step solution

1. Write down Gauss' Law for electric field

Gauss' Law states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space \((\epsilon_0)\), which is given by: \(\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}\).

2. Apply Gauss' Law to find the electric field outside the sphere

Consider a Gaussian surface which is a sphere of radius \(r (r \geq R)\) centered at the origin. As the surface is symmetric, the electric field at a point on the surface will be: \(\vec{E} = E \hat{r}\). The electric flux through the Gaussian surface is given by: \(\oint \vec{E} \cdot d\vec{A} = E \cdot 4\pi r^2\). According to Gauss' Law, \(\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}\). Therefore, \(E \cdot 4\pi r^2 = \frac{Q}{\epsilon_0}\). Solving for E, we get: \(E = \frac{Q}{4\pi\epsilon_0 r^2}\).

3. Find the electric field inside the sphere

For points inside the hollow sphere \((0\leq r<R)\), the enclosed charge is zero, and according to Gauss' Law, the electric field should be zero too, which means: \(E_{inside} = 0\).

4. Determine the electric potential inside and outside the sphere

The electric potential can be found by integrating the electric field along the path of interest. For points outside the sphere, integrate from infinity to r: \(V_{outside}(r) = -\int_{\infty}^{r} E_{outside} dr = -\int_{\infty}^{r} \frac{Q}{4\pi\epsilon_0 r'^2} dr' = \frac{kQ}{r}\). For points inside the sphere, the electric field is zero, so the electric potential remains constant: \(V_{inside}(r) = V_{outside}(R) = \frac{kQ}{R}\).

5. Find the electric potential at the center of the sphere

Finally, substitute \(r=0\) (the center of the sphere) in the formula \(V_{inside}(r) = \frac{kQ}{R}\): \(V(0) = \frac{kQ}{R}\). Comparing with the given options, we find that the correct answer is (c) \(kQ/R\).