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Chapter 23: Problem 6

A solid conducting sphere of radius \(R\) is centered about the origin of an \(x y z\) -coordinate system. A total charge \(Q\) is distributed uniformly on 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 conducting sphere? a) zero c) \(Q / 2 \pi \epsilon_{0} R\) b) \(Q / \epsilon_{0} R\) d) \(Q / 4 \pi \epsilon_{0} R\)

Short Answer

Expert verified
Answer: The electric potential at the center of the conducting sphere is Q / 4πε₀R.

Step by step solution

01

Understanding the electric field within a conducting sphere

The electric field within a conducting sphere is zero, because the charges reside on the surface and rearrange themselves to cancel any internal field. So there is no electric field inside the conducting sphere.
02

Calculate the electric potential using electric field

Since the electric field inside the sphere is zero, the electric potential is constant throughout the sphere. Recall that electric potential, V, is given by the relation: \(V = -\int \vec{E} \cdot d\vec{r}\) Since \(\vec{E}\) is zero inside the sphere, the integral yields a constant value.
03

Determine the reference point for the electric potential

We were given that the electric potential is zero at an infinite distance. Therefore, we can find the value of the constant potential inside the sphere by evaluating the potential on the surface: \(V(R) = k_e \frac{Q}{R}\) where \(k_e\) is the electrostatic constant, and equals to \(\frac{1}{4\pi\epsilon_0}\).
04

Electric potential at the center

Since the electric potential is constant throughout the sphere, the electric potential at the center of the sphere will be the same as the electric potential on the surface: \(V(0) = V(R) = k_e \frac{Q}{R} = \frac{Q}{4\pi\epsilon_{0}R}\) The electric potential at the center of the conducting sphere is \(Q / 4 \pi \epsilon_{0} R\). Therefore, the answer is option d.

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