7.3: Coulomb Potential

Learning Objectives

• Adapt the Coulomb Potential to discrete charge distributions using summation notation.
• Use the integral form of the Coulomb Potential to determine the electric potential from continuous charge distributions using single integrals.

Review

Using the relationship between electric fields and electric potential, we previously found an equation for the electric potential from a point charge when our zero-point reference was at infinity:

\[ V_{\text{point charge}}=\frac{1}{4\pi\epsilon_0}\frac{Q}{r} \]

Model

Explore

1. How many charges are shown? If they all have the same charge and the total charge of the line is \(Q\), what would the charge of each circle be (\(\Delta Q\))?
2. Label the charges in some way (for example, assign numbers or letters).
3. Draw a separation distance (scalar, not vector) from each charge to the point on the \(y\)-axis that represents the point of interest. Give each distance the same subscript label as its corresponding charge.

Critical Thinking & Concept Invention

1. Without performing calculations, discuss how you would determine the total electric potential from the distribution.
2. Express this method using summation notation:

   \[ V=\sum \]

3. Taking the limit that converts \(\Delta\) quantities into differentials and summations into integrals, rewrite the expression:

   \[ V=\int \]

Apply

Determine the electric potential at point \(P\) for the collection of charges shown. Express the result in summation notation using the equation developed for a discrete collection of charges.

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Summary

Just as any charge distribution can be viewed as a collection of point charges when calculating electric fields, the same approach applies to electric potential. This leads to the Coulomb Potential integral:

\[ V=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r} \]

Differential charge elements may be identified from charge densities such as \(\lambda\), \(\sigma\), and \(\rho\). For a general volume distribution:

\[ V=\frac{1}{4\pi\epsilon_0}\iiint\frac{\rho\,dV}{r} \]

Electric potential calculations are often simpler than electric field calculations because they involve scalar quantities rather than vectors. Once the potential is known, the electric field can be obtained from:

\[ \vec{E}=-\vec{\nabla}V \]

Model

A line of charge of length \(L\) with constant charge density has total charge \(Q\). The electric potential is evaluated at a point directly above the center of the line.

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. Since \(dl\) is a differential length along the object, identify which variable represents \(dl\) in this model.
2. Determine the linear charge density \(\lambda\) assuming it is constant.
3. Write a scalar expression for the separation distance using the coordinates and variables shown. Determine whether this distance depends on the position of the differential charge element.
4. Identify the coordinates corresponding to:
   • The top of the rod.
   • The middle of the rod.
   • The bottom of the rod.

Critical Thinking & Concept Invention

1. Determine the differential charge element \(dq\). Recall that for a one-dimensional object with constant charge density:

\[ dq=\lambda\,dl \]

2. Determine the limits of integration needed to integrate over the entire line.
3. Use the Coulomb Potential equation to determine the electric potential a distance \(x\) from the center of the line. Constants may be factored outside the integral.

\[ V=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r} \]

\[ V=\int d \]

Apply

Determine the electric potential at a point located a distance \(L/4\) below the end of the charged line shown in the accompanying figure.

Model

A ring has total charge \(Q\). The electric potential is evaluated directly above the center of the ring.

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. For a one-dimensional object, express the differential length \(dl\) in terms of the variables provided.
2. For constant linear charge density, determine:

\[ \lambda=\frac{Q}{L} \]

3. Determine \(dq\) for the ring and verify that it equals \(\lambda\,dl\).
4. Write a scalar expression for the separation distance between the differential charge element and the observation point. Determine whether this distance depends on the location of the differential element.

Critical Thinking & Concept Invention

1. Determine the limits of integration needed to integrate around the complete ring.
2. Use the Coulomb Potential equation to determine the electric potential a distance \(z\) above the center of the ring. Constants may be factored outside the integral.

\[ V=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r} \]

\[ V=\int d \]

Model

A disk has total charge \(Q\). The electric potential is evaluated directly above the center of the disk.

Image Credit: OpenStax University Physics Vol 2 (https://openstax.org/details/books/university-physics-volume-2)

Explore

1. For a two-dimensional object, determine the differential area \(dA\) of the ring-shaped element in terms of the variables provided.
2. For constant surface charge density, determine:

\[ \sigma=\frac{Q}{A} \]

3. Write a scalar expression for the separation distance between the differential area element and the observation point. Determine whether this distance depends on position.

Critical Thinking & Concept Invention

The electric potential for a charged ring has already been determined. Use that result with the substitution \(Q_{\text{ring}}\rightarrow dq\).

1. Determine \(dq\) for the disk and identify whether it depends on the location of the differential element.
2. Determine the limits of integration required to integrate over the entire disk.
3. Use the Coulomb Potential equation to determine the electric potential a distance \(z\) above the center of the disk. Constants may be factored outside the integral.

\[ V=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r} \]

\[ V=\int d \]