Defining Electric Potential and Electric Field

Let's begin with the definition of the change in electric potential, which essentially represents the work done per unit charge. This work is computed as a path integral of the force, specifically the electric force acting on a charge, leading to the following expression for the change in potential between points A and B.

Since the change in potential is derived from the electric field, it follows that the electric field is the derivative of the electric potential. However, a simple derivative does not yield a scalar value. This is where the del operator becomes essential, acting as a three-dimensional operator comprising derivatives.

Thus, the electric field at a certain point can be expressed as follows:

However, you cannot simply derive the electric potential from a singular value. It’s akin to trying to determine the slope at a single point—impossible. Therefore, you have two choices: First, if you know the electric potential as a function of x, y, and z, you can compute the derivatives conventionally. Alternatively, if you only possess numerical values for the electric potential, you can calculate a numerical derivative, which is the method I intend to use.

Finding the Electric Potential

Before utilizing the electric potential, you need to gather some values. Let’s assume there are two electric charges situated along the x-axis. Charge 1, valued at 6 nC, is located at the origin, while Charge 2, at -2 nC, is positioned at x = 0.01 meters. The selection of these values is largely arbitrary. Here’s a visual representation of the setup.

Next, we will calculate the electric potential at another point along the x-axis, specifically at x = 0.02 meters. The first step involves using the formula for electric potential from a single point charge, referenced from infinity.

In this equation, k represents the Coulomb constant (9 x 10^-9 Nm²/C²), and r indicates the distance from the charge to the point where the potential is being calculated. Given that there are two charges, you’ll compute the potential twice and sum the results.

That serves as a preliminary step. Next, we can employ this information to derive the electric field.

Calculating the Electric Field from the Electric Potential

To derive a numerical derivative, we'll follow this approach—keeping it straightforward by focusing solely on the x-direction. Begin by determining the electric potential at a specific point (designated as x0). Then, advance a small distance (referred to as dx) and assess the new electric potential. Finally, retreat by the same distance dx and measure the potential again. The x-component of the electric field will then be represented as:

Let’s proceed with this calculation. Utilizing Python for this computation simplifies the process significantly. Here’s the code for reference.

It's important to note that I employed the change in electric potential to compute the electric field's x-component. Additionally, I calculated the electric field through the principle of superposition using the two point charges. The results were remarkably similar, indicating a successful approach.

Once you’ve determined the x-component of the electric field at one point, you can replicate this calculation for any point along the axis, as illustrated below (the corresponding code is also provided).

Notice how the electric field derived from the superposition of individual charges aligns closely with the field calculated from the electric potential—a promising result.

Consider this: At what position along the x-axis does the x-component of the electric field equal zero? Observations suggest it occurs near x = 0.0236 meters (as indicated by the graph). However, is the electric potential at this point also zero? Not at all. Remember that the electric field is influenced by changes in electric potential, not by its absolute value. It is indeed common for the electric field to be zero while the electric potential remains a non-zero quantity.

Homework Assignment

As an exercise, try to calculate the vector components of the electric field at a location that does not lie along the x-axis. This will require you to compute numerical derivatives in both the x and y directions. Enjoy the challenge!

This video explains how to derive the electric field from electric potential using the fundamental principles of physics.

In this video, learn how to find the vector electric field created by two point charges and the principles behind their interactions.