7.4 Determining Field from Potential

Recall that we were able, in certain systems, to calculate the potential by integrating over the electric field. As you may already suspect, this means that we may calculate the electric field by taking derivatives of the potential, although going from a scalar to a vector quantity introduces some interesting wrinkles. We frequently need $\vec{E}$ to calculate the force in a system; since it is often simpler to calculate the potential directly, there are systems in which it is useful to calculate V and then derive $\vec{E}$ from it.

In general, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of $\vec{E}$ and also in the direction of lower potential V. Furthermore, the magnitude of $\vec{E}$ equals the rate of decrease of V with distance. The faster V decreases over distance, the greater the electric field. This gives us the following result.

For continually changing potentials, $Δ V$ and $Δ s$ become infinitesimals, and we need differential calculus to determine the electric field. As shown in Figure 7.27, if we treat the distance $Δ s$ as very small so that the electric field is essentially constant over it, we find that

$E_{s} = - \frac{d V}{d s} .$

The figure shows the electric field component of two points A and B separated by distance delta s and having a potential difference of delta V. — **Figure 7.27** The electric field component, $E sub 1$ , along the displacement $Δ s$ is given by $E = - \frac{Δ V}{Δ s}$ . Note that A and B are assumed to be so close together that the field is constant along $Δ s$ .

Therefore, the electric field components in the Cartesian directions are given by

$E_{x} = - \frac{\partial V}{\partial x}, E_{y} = - \frac{\partial V}{\partial y}, E_{z} = - \frac{\partial V}{\partial z} .$

This allows us to define the “grad” or “del” vector operator, which allows us to compute the gradient in one step. In Cartesian coordinates, it takes the form

$\vec{\nabla} = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z} .$

With this notation, we can calculate the electric field from the potential with

$E → equals − ∇ → V,$

a process we call calculating the gradient of the potential.

If we have a system with either cylindrical or spherical symmetry, we only need to use the del operator in the appropriate coordinates:

$Cylindrical: \vec{\nabla} = \hat{r} \frac{\partial}{\partial r} + \hat{φ} \frac{1}{r} \frac{\partial}{\partial φ} + \hat{z} \frac{\partial}{\partial z}$

$Spherical: \vec{\nabla} = \hat{r} \frac{\partial}{\partial r} + \hat{θ} \frac{1}{r} \frac{\partial}{\partial θ} + \hat{φ} \frac{1}{r sin θ} \frac{\partial}{\partial φ}$

Example 1 Electric Field of a Point Charge

Calculate the electric field of a point charge from the potential.

Strategy The potential is known to be $V = k \frac{q}{r}$ , which has a spherical symmetry. Therefore, we use the spherical del operator in the formula $E → equals − ∇ → V$ .

Solution Performing this calculation gives us

$\vec{E} = − (\hat{r} \frac{\partial}{\partial r} + \hat{θ} \frac{1}{r} \frac{\partial}{\partial θ} + \hat{φ} \frac{1}{r sin θ} \frac{\partial}{\partial φ}) k \frac{q}{r} = − k q (\hat{r} \frac{\partial}{\partial r} \frac{1}{r} + \hat{θ} \frac{1}{r} \frac{\partial}{\partial θ} \frac{1}{r} + \hat{φ} \frac{1}{r sin θ} \frac{\partial}{\partial φ} \frac{1}{r}) .$

This equation simplifies to

$\vec{E} = − k q (\hat{r} \frac{−1}{r^{2}} + \hat{θ} 0 + \hat{φ} 0) = k \frac{q}{r^{2}} \hat{r}$

as expected.

Significance We not only obtained the equation for the electric field of a point particle that we’ve seen before, we also have a demonstration that $\vec{E}$ points in the direction of decreasing potential, as shown in Figure 7.28.

Figure 7.28 Electric field vectors inside and outside a uniformly charged sphere.

Summary

Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field.
This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator.

Conceptual Questions

If the electric field is zero throughout a region, must the electric potential also be zero in that region?

No. It will be constant, but not necessarily zero.

Explain why knowledge of $\vec{E} (x, y, z)$ is not sufficient to determine V(x,y,z). What about the other way around?

Problems

Throughout a region, equipotential surfaces are given by $z = constant$ . The surfaces are equally spaced with $V equals 100 V$ for $z = 0.00 m, V = 200 V$ for $z = 0.50 m, V = 300 V$ for $z equals 1.00 m.$ What is the electric field in this region?

The problem is describing a uniform field, so $E equals 200 V/m$ in the –z-direction.

In a particular region, the electric potential is given by $V = − x y^{2} z + 4 x y .$ What is the electric field in this region?

Calculate the electric field of an infinite line charge, throughout space.

Apply $E → equals − ∇ → V$ with $\vec{\nabla} = \hat{r} \frac{\partial}{\partial r} + \hat{φ} \frac{1}{r} \frac{\partial}{\partial φ} + \hat{z} \frac{\partial}{\partial z}$ to the potential calculated earlier,
$V = −2 k λ ln s : \vec{E} = 2 k λ \frac{1}{r} \hat{r}$ as expected.