Unlocking Electric Potential- A Comprehensive Guide to Deriving Electric Potential from Electric Field

How to Find Electric Potential from Electric Field

Electric potential and electric field are two fundamental concepts in electromagnetism. The electric field represents the force experienced by a charged particle in an electric field, while the electric potential measures the amount of work done to move a unit positive charge from one point to another. In this article, we will discuss how to find electric potential from electric field, providing a step-by-step guide to help you understand the relationship between these two quantities.

Understanding the Relationship

The relationship between electric potential (V) and electric field (E) is given by the equation:

V = -∫E·dr

This equation shows that the electric potential at a point in space is equal to the negative integral of the electric field along a path from a reference point to the point of interest. The negative sign indicates that the electric potential decreases as the electric field increases.

Step-by-Step Guide

1. Choose a reference point: Select a point in space to serve as the reference point for calculating the electric potential. This point can be any location, but it is often chosen to be at infinity or at the origin.

2. Determine the electric field: Find the electric field at the point of interest. This can be done using Coulomb’s law or Gauss’s law, depending on the situation.

3. Calculate the displacement vector: Determine the displacement vector (dr) from the reference point to the point of interest. This vector represents the direction and magnitude of the path taken to move the unit positive charge.

4. Evaluate the dot product: Calculate the dot product of the electric field vector and the displacement vector (E·dr). The dot product gives the magnitude of the work done by the electric field along the path.

5. Integrate the dot product: Integrate the dot product over the path from the reference point to the point of interest. This gives the total work done by the electric field on the unit positive charge.

6. Negate the integral: Multiply the result of the integral by -1 to obtain the electric potential at the point of interest.

Example

Suppose we have a point charge Q located at the origin, and we want to find the electric potential at a point P located at a distance r from the origin along the x-axis. The electric field at point P is given by:

E = kQ/r^2 (i^)

where k is the Coulomb constant, Q is the charge, r is the distance from the origin, and i^ is the unit vector along the x-axis.

The displacement vector from the origin to point P is:

dr = r i^

Now, we can calculate the dot product:

E·dr = (kQ/r^2 i^) · (r i^) = kQ/r

Integrating this over the path from the origin to point P gives:

V = -∫E·dr = -∫(kQ/r)dr = -kQ ln(r)

The electric potential at point P is thus:

V = -kQ ln(r)

This example demonstrates how to find the electric potential from the electric field using the integral method. By following the steps outlined in this article, you can calculate the electric potential at any point in space given the electric field.