Question

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

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

Step-by-step solution

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.

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.

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 \(\frac{1}{4\pi\epsilon_0}\).

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.

Key concepts

Conducting Sphere

A conducting sphere is a fascinating concept in physics because it demonstrates how charges behave. When we talk about a conducting sphere, we are referring to an object that can distribute charge over its surface evenly. When a charge is placed on a conducting sphere, the charge spreads out uniformly over its outer surface. This is because like charges repel, and they try to stay as far away from each other as possible.

Inside the sphere, there is one interesting property: the electric field is zero. This happens because the charges on the surface create an internal field that cancels itself out. That's why the electric field inside a conducting sphere is always zero, no matter how much charge there is on the outside. This property ensures that the electric potential inside the sphere remains constant. All the points inside the conducting sphere, therefore, have the same electric potential.

Electrostatics

Electrostatics is the branch of physics that deals with the study of forces, fields, and potentials arising from static charges. It helps us understand how electric charges interact when they are at rest in a given space.

In electrostatics, one key principle is that charges reside on the surface of conductors in equilibrium. This means the electric field inside a conductor is zero, resulting in a constant electric potential throughout its volume. Understanding this helps us compute the potential in various scenarios, like in a conducting sphere.

Consider an isolated system where we need to calculate the electric potential. The potential at any point is defined by integrating the electric field along a path, as given by the equation:

• \(V = -\int \vec{E} \cdot d\vec{r}\)

However, if the electric field \(\vec{E}\) is zero inside a sphere, the potential is simply determined by known values at specific boundaries. This makes it straightforward to find potential anywhere inside the conductor.

Electric Field

The electric field is a vector field around a charged particle that represents the force exerted per unit positive charge at any given point. It tells us about the influence a charge has on the surrounding space. In the context of a conducting sphere, the behavior of the electric field is quite intriguing.

Since all the charge on a conducting sphere resides on its surface, the electric field inside the sphere is zero. This lack of an internal electric field results from the mutual repulsion of like charges. Outside the sphere, the electric field diminishes as you move further away, just like with a point charge.

The strength of this electric field just outside the surface of the sphere can be calculated using Gauss's Law. The formula for this is:

• \(E = k_e \frac{Q}{R^2}\)

Where \(k_e\) is Coulomb's constant. Understanding these properties about electric fields helps us comprehend concepts like electric potential and electric forces in spherical conductors.