Question

A solid metal ball with a radius of \(3.00 \mathrm{~m}\) has a charge of \(4.00 \mathrm{mC}\) If the electric potential is zero far away from the ball, what is the electric potential at each of the following positions? a) at \(r=0 \mathrm{~m},\) the center of the ball b) at \(r=3.00 \mathrm{~m},\) on the surface of the ball c) at \(r=5.00 \mathrm{~m}\)

Short answer

Answer: The electric potentials are \(1.20 \times 10^7 \mathrm{~V}\) at the center of the ball (r=0), \(1.20 \times 10^7 \mathrm{~V}\) at the surface of the ball (r=3.00 m), and \(7.19 \times 10^6 \mathrm{~V}\) at a distance of 5.00 m from the ball.

Step-by-step solution

Recall the formula for electric potential due to a point charge

The formula for the electric potential V at a distance r from a point charge Q is given by: \(V = \dfrac{kQ}{r}\) Here, k is the Coulomb's constant (\(8.99 \times 10^9 \mathrm{~N m^2 C^{-2}}\)) and Q is the charge.

Calculate the potential at the center of the ball (r=0)

When we are at the center of the metal ball (r=0), the potential at the surface of the ball will be constant throughout the ball. So we can calculate the potential at the surface and that will be the same at the center.

Calculate the potential at the surface (r=3.00 m)

To find the potential at the surface of the ball, we can use the formula for the potential due to a point charge. Plug in the given values (Q = \(4.00 \mathrm{mC}\) and r = \(3.00 \mathrm{~m}\)): \(V = \dfrac{kQ}{r} = \dfrac{8.99 \times 10^9 \mathrm{~N m^2 C^{-2}} \times 4.00 \times 10^{-3} \mathrm{C}}{3.00 \mathrm{~m}} = 1.20 \times 10^7 \mathrm{~V}\) Thus, the potential at the surface of the ball is \(1.20 \times 10^7 \mathrm{~V}\), and it is the same as the potential at the center of the ball (r = 0).

Calculate the potential at r=5.00 m

To find the potential at a distance r = 5.00 m from the metal ball, we can again use the formula for the potential due to a point charge. Plug in the given values (Q = \(4.00 \mathrm{mC}\) and r = \(5.00 \mathrm{~m}\)): \(V = \dfrac{kQ}{r} = \dfrac{8.99 \times 10^9 \mathrm{~N m^2 C^{-2}} \times 4.00 \times 10^{-3} \mathrm{C}}{5.00 \mathrm{~m}} = 7.19 \times 10^6 \mathrm{~V}\) Thus, the electric potential at a distance of 5.00 m from the ball is \(7.19 \times 10^6 \mathrm{~V}\).

Key concepts

Electric Potential Formula

The electric potential formula is critical for understanding how the potential energy of a charged particle changes due to its position in an electric field. Simply put, electric potential at a point in space is defined as the work done by an external agent in bringing a unit positive charge from infinity to that point, without any acceleration. The formula for electric potential (V) due to a point charge (Q) is given by:

\[V = \frac{kQ}{r}\]

Here, V is the electric potential, Q is the charge, r is the distance from the charge to the point of interest, and k is the Coulomb's constant, which leads us to its importance in this equation.

Coulomb's Constant

Coulomb's constant (k), also known as the electric force constant, is a proportionality factor that appears in Coulomb's law as well as in the electric potential formula. It is a fundamental physical constant and has the value of approximately \[8.99 \times 10^9 \mathrm{~N m^2 C^{-2}}\].

The constant represents the electric force between two charges of one Coulomb each separated by a distance of one meter. It allows us to calculate the electric potential, as well as forces in electrostatics. Since it is a constant, its value is crucial for calculating accurate electric potentials, reinforcing the interconnectedness between a charge's potential and the forces within an electric field.

Charge Distribution

Charge distribution refers to how charge is spread over an object. In electrostatics, objects can be charged in a uniform or non-uniform manner. For spherical objects like a solid metal ball, the charge is often assumed to be uniformly distributed over its surface.

Uniform vs. Non-uniform Charge Distribution

Uniform charge distribution occurs when the charge is spread evenly across an object's surface, leading to a symmetric electric field outside the object. This is contrasted with non-uniform charge distribution, which can create irregular electric fields. The concept of charge distribution is fundamental in calculating electric potentials, especially when dealing with complex geometries or different materials.