Calculating the Field from the Potential

Calculating the Electric Field from the Electric Potential

In electrostatics, the electric field $\vec{E}$ and electric potential $V$ are closely related. The electric field is the negative gradient of the electric potential. This relationship allows us to calculate the electric field if we know the electric potential, and vice versa.

1. General Relationship Between Electric Field and Potential

The electric field $\vec{E}$ at a point is the negative gradient of the electric potential $V$. In mathematical terms, this is:

$$\vec{E} = - \nabla V$$

This equation expresses that the electric field points in the direction of greatest decrease of the potential, and the magnitude of the field is proportional to how rapidly the potential changes with distance.

Components of the Gradient in Cartesian Coordinates:

If the potential $V$ is a function of the spatial coordinates $x$, $y$, and $z$, the gradient in Cartesian coordinates is given by:

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

Therefore, the electric field components are:

$$\vec{E} = - \hat{i} \frac{\partial V}{\partial x} - \hat{j} \frac{\partial V}{\partial y} - \hat{k} \frac{\partial V}{\partial z}$$

In Spherical Coordinates:

In spherical coordinates ($r$, $\theta$, $\phi$), where the potential $V$ depends on the radial distance $r$, the polar angle $\theta$, and the azimuthal angle $\phi$, the gradient is:

$$\nabla V = \hat{r} \frac{\partial V}{\partial r} + \hat{\theta} \frac{1}{r} \frac{\partial V}{\partial \theta} + \hat{\phi} \frac{1}{r \sin \theta} \frac{\partial V}{\partial \phi}$$

So, the electric field in spherical coordinates is:

$$\vec{E} = - \hat{r} \frac{\partial V}{\partial r} - \hat{\theta} \frac{1}{r} \frac{\partial V}{\partial \theta} - \hat{\phi} \frac{1}{r \sin \theta} \frac{\partial V}{\partial \phi}$$

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2. Calculating Electric Field from Potential: Examples

Now, let's look at how to calculate the electric field from the potential in specific situations:

a. Electric Field Due to a Point Charge

The potential due to a point charge $Q$ at a distance $r$ from the charge is:

$$V(r) = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r}$$

To find the electric field, we take the negative gradient of the potential. In spherical coordinates, the gradient of $V(r)$ with respect to $r$ is:

$$\frac{\partial V}{\partial r} = - \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$$

So, the electric field $\vec{E}$ is:

$$\vec{E} = - \hat{r} \left( - \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \right)$$

Thus, the electric field due to a point charge is:

$$\vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \hat{r}$$

This is Coulomb's law, which states that the electric field due to a point charge is radially outward (for a positive charge) or inward (for a negative charge) and decreases with the square of the distance.

b. Electric Field Due to a Uniformly Charged Infinite Line

For a uniformly charged infinite line with linear charge density $\lambda$, the potential at a distance $r$ from the line is:

$$V(r) = \frac{\lambda}{2 \pi \epsilon_0} \ln \left( \frac{1}{r} \right)$$

To find the electric field, we take the gradient of $V(r)$ with respect to $r$:

$$\frac{\partial V}{\partial r} = - \frac{\lambda}{2 \pi \epsilon_0 r}$$

The electric field is then:

$$\vec{E} = - \hat{r} \left( - \frac{\lambda}{2 \pi \epsilon_0 r} \right)$$

Thus, the electric field due to an infinite line of charge is:

$$\vec{E} = \frac{\lambda}{2 \pi \epsilon_0 r} \hat{r}$$

This electric field points radially outward from the line if $\lambda > 0$ and inward if $\lambda < 0$, and it decreases with distance from the line.

c. Electric Field Due to a Uniformly Charged Disk

The potential at a point along the axis of a uniformly charged disk with surface charge density $\sigma$ and radius $R$ is:

$$V(z) = \frac{\sigma}{2 \epsilon_0} \left( \sqrt{R^2 + z^2} - z \right)$$

To find the electric field along the axis, we take the derivative of the potential $V(z)$ with respect to $z$:

$$\frac{dV}{dz} = \frac{\sigma}{2 \epsilon_0} \left( \frac{z}{\sqrt{R^2 + z^2}} - 1 \right)$$

So, the electric field along the axis of the disk is:

$$E(z) = - \frac{dV}{dz} = \frac{\sigma}{2 \epsilon_0} \left( 1 - \frac{z}{\sqrt{R^2 + z^2}} \right)$$

This shows that the electric field on the axis of a uniformly charged disk decreases as $z$ increases and approaches a value similar to the field of a point charge at large $z$.

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3. Electric Field from Potential for More Complex Distributions

For more complex charge distributions, the general process for calculating the electric field from the potential involves:

1. Writing down the expression for the potential: This could be due to a point charge, a charged sphere, a charged line, or any other configuration of charges.
2. Taking the gradient: You compute the gradient of the potential to find the electric field.
3. Interpreting the result: The resulting electric field may be in vector form or expressed in terms of radial, angular, or Cartesian components.

For example, in the case of a spherically symmetric charge distribution, you would calculate the electric potential for that distribution, and the electric field can be found by taking the derivative of that potential with respect to $r$.

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4. Summary

• The electric field $\vec{E}$ can be calculated from the electric potential $V$ using the relationship: $$\vec{E} = - \nabla V$$
• The gradient operation tells us how the potential changes in space, and the electric field points in the direction of greatest decrease of the potential.
• In Cartesian coordinates, the electric field components are found by taking partial derivatives of the potential with respect to $x$, $y$, and $z$.
• In spherical coordinates, the gradient is expressed in terms of $r$, $\theta$, and $\phi$, and the electric field is calculated accordingly.
• Examples: For a point charge, the electric field is $\frac{Q}{r^2}$; for an infinite line charge, it is $\frac{\lambda}{r}$; and for a uniformly charged disk, it can be computed along the axis using the derivative of the potential.

This relationship between electric field and potential is a fundamental concept in electrostatics and is widely used in solving problems related to electric fields and potentials.