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AP Physics 2: Exam Prep26 chapters | 141 lessons

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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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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).

Using the Pythagorean theorem, we get the following magnitude of the net electric field at location X.

Vector quantities need a direction along with a magnitude. Using trigonometry, we can get the angle θ

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

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 line(s) 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.

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AP Physics 2: Exam Prep26 chapters | 141 lessons

- Electric Fields: Definition & Examples 6:09
- Electric Force: Definition & Equation 6:18
- Electric Field & the Movement of Charge 6:41
- Strength of an Electric Field & Coulomb's Law 6:46
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- Electric Potential Energy: Definition & Formula 4:19
- Finding the Electric Potential Difference Between Two Points 5:31
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