24.10:
Determining Electric Field From Electric Potential
For a system of charged particles, the integral of the dot product of the electric field with the displacement of a test charge gives the difference in potential between the initial and final positions.
This change in potential will be infinitesimally small for an infinitesimally smaller displacement of a test charge.
Simplifying and expressing the above equation in component form gives the electric field as a partial derivative of the change in potential along a particular axis.
In vector notation, the electric field is expressed as a gradient of the electric potential. Here, the operator is called a del operator.
At each point in space, del operators give the electric field pointing in the direction along which the steepest decrease in the potential is present.
For a system with spherical or cylindrical symmetry, an appropriate del operator can be used.
For example, the electric potential of a positive point charge has spherical symmetry, and the corresponding electric field can be calculated using a del operator in spherical symmetry.
24.10:
Determining Electric Field From Electric Potential
The electric field and electric potential are related to each other. If the electric field at various points in the region of interest is known, it can be used to calculate the electric potential difference between any two points. Similarly, if the electric potential is known for various points, then it is possible to calculate the electric field.
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 the electric field and the direction of the lower electric potential. Furthermore, the magnitude of the electric field equals the rate of decrease in the electric potential with distance. The faster the potential decreases over a distance, the greater the electric field. The electric field is expressed as a gradient of the electric potential in the equation form.
At each point, the potential gradient always points in the direction in which the potential increases the most with an infinitesimal change in position. Due to the negative sign in the above equation, the electric field points in the direction where the electric potential decreases the most. The above equation does not demand a particular zero point where the potential is zero, but instead this is an arbitrary point. The potential gradient, irrespective of where the zero has been defined, would always be the same.
Depending on the system's symmetry, the gradient operator can be used in different geometries. For example, the electric field of a point charge can be calculated using spherical symmetry.
Suggested Reading
- Young, H.D and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 23.5; pages 774-775.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Section 7.4; pages 316–318. Retrieved from https://openstax.org/details/books/university-physics-volume-2