Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering
Coulomb's law can be used to calculate the electric force between two point charges, but what do you do when there are more than two charges present? In this lesson, you'll learn how to calculate the magnitude and direction of the electric force when multiple charges are present.
Magnitude & Direction
When you hold a ball up above the ground and let it go, you know that it will fall toward the earth. This happens because the earth exerts an attractive force on the ball, causing it to move toward the earth, speeding up as it gets closer.
Something similar happens when you hold two charges near each other and let them go. The charges will exert electric forces on each other, causing them to either move apart or come together. However, unlike the gravitational force exerted by the earth on the ball, which is always an attractive force, the electric force between two point charges can be either attractive or repulsive, depending on the type of charges involved.
If the two charges are different, with one being positive and one negative, then they will attract each other. But, if the two charges are the same, with both being either positive or negative, then they will repel each other.
Coulomb's Law says that the magnitude of the electric force between two charged objects is directly proportional to the charge on each object (symbolized by q1 and q2) and inversely proportional to the distance between the charges (r). Therefore, you can use the following equation to calculate the magnitude of the force between ANY two charged objects:
Remember that this will only give you the magnitude of the force, and not the direction, so you should use the absolute value of q1 and q2. The magnitude of a force will never be a negative number!
Two Point Charges
Let's first try to use Coulomb's Law to calculate the magnitude and direction of the electric force on a point charge when there are only two point charges present.
In the situation shown here, what is the magnitude and direction of the force exerted on the charge on the left by the charge on the right?
First, determine the direction of the force on q1. Since one charge is negative and the other is positive, the charges will exert attractive forces on each other. Therefore, the electric force on q1 will be directed toward the right.
Next, use Coulomb's Law to calculate the magnitude of the force, like this:
Therefore, the electric force exerted on q1 is 1.5x10-6 N directed towards the right.
Point Charges in a Line
Next, let's try to find the net force on a charge when more than one electric force is exerted on it, as shown here. What is the net force on q1 now?
In this situation, both q2 AND q3 exert electric forces on q1. To find the net force on q1, first determine the direction of the force that each of the other charges will exert on it. As before, q2 will exert an attractive force to the right, which we already calculated. Because q1 and q3 are both positively charged, then q3 will exert a repelling force on q1 that will be directed toward the left, like this:
Then, use Coulomb's Law to calculate the magnitude of each force. We already know the magnitude of F2 since we calculated it in the first example, so we just need to find F3:
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Finally, to find the net force, notice that F3 is directed to the left so it is in a negative direction, while F2 is directed to the right so it is in a positive direction. Therefore, find the net force by subtracting the magnitude of F3 from the magnitude of F2, like this:
The net force on F1 is directed toward the right and has a magnitude of 6.36x10-7 N.
Point Charges in a 2D Plane
Finally, let's look at how to find the net force when the charges aren't in a straight line. If q1, q2, and q3 are arranged as shown here, what is the net force on q1 now?
The distances between the charges are the same as they were when the charges were arranged in a line, so the magnitudes of the two individual electric forces have not changed. However, the magnitude and direction of the net force has definitely changed!
Once again, you need to first determine the direction of each electric force on q1. Remember that forces that are the same will repel each other, and forces that are different will attract each other.
To find the net force, however, you cannot simply add or subtract these forces. Forces are vectors, so you must add them together using vector algebra. To add two vectors, start by drawing the first vector in the correct direction, then draw the second vector from the tip of the first. The sum of the two vectors is a line drawn from the beginning of the first vector to the end of the second vector. This makes a triangle, so you can use trigonometry to find the magnitude and direction of the resultant vector.
To calculate the electric force on a point charge, first determine the direction of the force. Two charges that are the same will repel each other, while two charges that are different will attract each other. Then, use Coulomb's Law, which states that the magnitude of the electric force between two charged objects is directly proportional to the charge on each object, to find the magnitude of the electric force between any two charges. If there is more than one electric force exerted on a charge, find the net force by finding the vector sum of all the individual forces that act on the charge.
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