Electric Field Formula, Magnitude & Direction | Calculate the Magnitude of an Electric Field – Video & Lesson Transcript | Study.com

Video Transcript

The Electric Field

Any charged particle has an electric field associated with it, since electric fields are generated by charged particles. 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.

A point in space may have multiple electric fields stemming from multiple charged particles. It’s our job to calculate the net electric field at any point. It’s helpful to visualize the multiple fields by using a diagram, like this one appearing here:

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.

In this equation, 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.

Three Charge Electric Field Problem

Let’s work on a three-charge electric field problem. Here’s our scenario:

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.

Charge 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. This next figure 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 our first equation. One of the variables in this equation is the distance between the charge and point X. This next figure 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. This chart appearing shows all the data needed to calculate the magnitudes of the electric fields at point X.

Chart 1.

Using our first equation 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 can be seen here:

The net electric field determined shows the magnitude of the net field at location X in each respective direction. Let’s graphically represent the net electric field and determine the net electric field as a magnitude and a direction (θ).

This graph shows the relative magnitudes of the x and y components of the electric field. Location X is in the center of the grid, and the red arrow is the net electric field.

Using the Pythagorean theorem, we get the following magnitude of the net electric field at location X, which as we can see, ends up being:

Vector quantities need a direction along with a magnitude. Using trigonometry, we can get the angle θ, which as we can see, is 40° below the negative x-axis.

This angle justifies our graphical representation of the net electric field at point X, and our final answer is our two answers combined:

.

Simple as that!

Lesson Summary

All right, let’s take a moment or two to review. As we learned, electric fields are generated by charged particles. The magnitude of an electric field at any location relative to a single charged particle or system of multiple charged particles can be determined using the electric field strength equation.

To determine a net electric field at a specific point:

1. Draw a sketch of the field lines at the point where the net electric field is to be determined. Field lines point towards negative charges and away from positive charges.
2. Determine the magnitude of each field at that location using the electric field equation.
3. Resolve the magnitude of each field line into component directions.
4. Add component directions independently.
5. If you want your answer in magnitude direction notation use the Pythagorean theorem to get the magnitude of the net field, and use trigonometry to get the angle of the net field with respect to one of the axes.