How to Calculate the Magnitude of an Electric Field

The flow of charge carriers through a conducting material is called electricity.

An electric field is a force field surrounding electrically charged particles. Mathematically, the strength of an electric field at any given point is defined as F / q, where F is the force a particle experiences and q is its charge. Since like charges repel, all the net charge of a conductor is on its surface, so for nonpoint charges you can often use Gauss's Law to evaluate the strength of the electric field. The following steps show how to calculate this field for a charged sphere.

Things You'll Need

• Pencil
• Paper
• Calculator

Instructions

1. Imagine that you have a metal sphere of radius R with a charge Q on its surface; this charge is uniformly distributed. We want to find the strength of the electric field at a distance r from the surface of the sphere, so imagine this metal sphere is surrounded by a second sphere of radius r, where r is greater than R.

2. Write out the Gauss's Law equation: the area integral of the electric field vector is equal to the charge enclosed by a surface divided by a constant called the permittivity of free space.

3. Note that the electric field vector is everywhere perpendicular to any given surface element since you are dealing with a symmetrical surface. Consequently, you can simplify this equation to E times the surface area of the sphere with radius r. The surface area of a sphere is 4πr squared, so our equation is now E times 4πr squared = Q / permittivity constant.

4. Divide both sides by 4πr squared to obtain E = Q / permittivity times 4πr squared. This equation gives you the electric field at distance r.

Tips & Warnings

• Note that the equation found here is the same as the Coulomb's Law equation for a point charge. It follows from Gauss's Law that the electric field outside a conducting sphere is equal to the electric field around a point charge with a charge of equal magnitude.

• It also follows from Gauss's Law that the electric field inside a sphere of charge is 0.

References

• University of Tennessee Physics: Gauss' Law
• Georgia State University Hyperphysics: Electrical Field, Spherical Geometry
• Georgia State University Hyperphysics: Gauss' Law