Determining & Representing Magnitude & Direction of Electrical Fields

Instructor: Matthew Bergstresser
Electric fields are associated with charged particles. Multiple charges will generate multiple electric fields that can be vectorally added to determine the net electric field. This lesson will explain how to determine the net electric field resulting from multiple charged particles.

The Electric Field

Any charged particle has an electric field associated with it. Since there are two versions of electric charge (positive and negative) there are two types of electric fields. Positive charges produce electric fields that point away from the charge and end at infinity. The electric field associated with a negative charge starts at infinity and ends on the negatively charged particle.

Figure 1. Electric fields of positive and negative charges.

A point in space may have multiple electric fields stemming from multiple charged particles. It is our job to calculate the net electric field at any point. It is helpful to visualize the multiple fields by using a diagram.

Figure 2. The X marks the location where we might be asked to determine the net electric field.

Since electric fields are vectors we must do vector math to determine the net field at any location. Before doing the vector math, though, we must use the electric-field-strength equation to determine the magnitude of each charge's field at said location.

Equation 1. r is the distance from the charge to the point where the field strength is to be determined, and k is a constant telling us the unit for charge must be coulombs, and the unit for distance must be meters.

Let's work on a three-charge electric field problem.


Three charged particles exist on the circumference of a circle with radius R = 10 mm as shown in the diagram. Charge 1 (q1) is -1.5 µC, charge 2 (q2) is 2.9 µC, and charge 3 (q3) is 4.3 µC. Determine the net electric field directly across from q2 on the circumference of the circle at the point marked by the X.

Figure 3.

Charges q2, and q3 are positively charged so the electric field lines point away from the charges and we draw them pointing away from location X. Charge q1 is negatively charged so the electric field lines point at the charge. To represent this, we draw an arrow pointing at q1 from location X. Figure 4 shows these electric fields, and they are represented in different colors to match the charge that generates them.

Figure 4.

To determine the magnitude of these charges we need to use Equation 1. One of the variables in this equation is the distance between the charge and point X. Figure 5 shows the distances from each charge to point X.

Figure 5.

The distances between charges q1 and q3 are hypotenuses of right triangles with the right angle of the triangles located at the center of the circle. The Pythagorean theorem gives us √2 R as the distances between q1 and point X, and between q3 and point X. The distance between q2 and point X is 2R.

The other variable in the electric-field-strength equation is value of the charges themselves. Chart 1 shows all the data needed to calculate the magnitudes of the electric fields at point X.

Chart 1.

Using Equation 1 we can calculate the magnitude of each electric field. Since electric fields are vectors we will use these magnitudes along with the sine and cosine of the angle θ inside the triangles to determine the horizontal and vertical components of the electric field. The electric field stemming from q2 will not require any trigonometric calculations because it acts in the pure y-direction.

Each side of the E 1 triangle in Figure 5 has the same value which gives us the 45 o shown in the equations.


Each side of the E 3 triangle in Figure 5 has the same value which gives us the 45 o shown in the equations.

Now we can vectorially add these three electric fields being sure to only add x-components together and y-components together.


Our resultant electric field in unit-vector notation is


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