A system of charges can have electrostatic potential energy due to where each charge is located relative to the others. In this lesson, learn about electrostatic potential energy and how to calculate it.
Attraction and Repulsion of Charges
When you brush your hair on a cold day, why does it suddenly stand up? It looks like magic, but it's really caused by static electricity.
Understanding static electricity allows you to understand how different types of electric charges interact with each other. Charges on objects come from tiny subatomic particles called electrons and protons. There are only two types of electric charge: positive and negative. Electrons are negatively charged, while protons are positively charged.
These positive and negative charges exert forces on each other. We know that if two objects have the same type of charge, they will repel, or push each other away. If they have different charges (one positive and one negative), they will attract each other.
Now, let's look at what happened to your hair in more detail. When you brushed it, some electrons from the brush were transferred to your hair. These extra electrons gave each hair's strand a small negative charge. Because your hair is all negatively charged now, each strand repels the others until your hair stands on end!
Electrostatic Potential Energy
One way to measure the effects of these types of interactions between charges is to calculate the electrostatic potential energy of a system of charges. In general, potential energy is any kind of energy that is stored within a system. As this stored energy turns into kinetic energy, the object will start to move and will keep speeding up until all the potential energy has become kinetic energy.
A pair of charges will always have some potential energy because if they are released from rest, they will either start moving towards (if the charges are different) or away (if the charges are the same) from each other. Electrostatic potential energy is specifically the energy associated with a set of charges arranged in a certain configuration.
The potential energy (Ue) depends on the amount of charge that each object contains (q), how far apart the charges are (r), and Coulomb's constant (k):
Let's look at a couple of examples of how to calculate the electrostatic potential energy of a pair of charges. In all of these problems, remember that charge is measured in units of Coulombs (C) and energy is measured in units of Joules (J).
Same Type of Charge
Let's first look at an example that covers the same type of charge. Two small spheres, both with a charge of +0.005 C, are held 0.25 meters apart from each other. What is the electrostatic potential energy of this system of charges?
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Did you notice that the potential energy was positive? When the charges are the same, the electrostatic potential energy of the system will always be positive. This is because these two spheres really want to get away from each other. They repel each other more and more the closer they get. Therefore, it requires energy to be put into the system in order to move them into this position. That energy becomes stored as potential energy. If you released the two spheres at this point, they would immediately fly apart as the stored potential energy was converted into kinetic energy.
What would happen if the spheres were moved to 0.5 m apart? How would the electrostatic potential energy change? Because the potential energy is inversely proportional to the distance between the charges, this would make the potential energy decrease by half.
Now, let's look at an example that has different types of charge. Now, let's replace one of the spheres with a sphere that has -0.001 C of charge. If the spheres are once again held 0.25 meters apart, what is the electrostatic potential energy of this new charge system?
The potential energy of this system of charges was negative! This is because one of the charges was positive and the other was negative. Whenever you have a system of charges that contains two objects with DIFFERENT charges, then the potential energy of the system will always be negative.
Because objects with different types of charge attract each other, it doesn't require any input of energy to push these two spheres together. Instead, it requires energy to hold them apart, so as they get closer and closer, the amount of stored potential energy decreases. If electrostatic potential energy is assumed to be zero when they are really, really far apart, then the potential energy would be negative when they are close together like this.
Potential energy is energy that is stored in a system that can make objects in the system move. Electrostatic potential energy is specifically the energy associated with a set of charges arranged in a certain configuration. It depends on the amount of charge that each object contains as well as how far apart the charges are.
You can calculate the electrostatic potential energy of a pair of charges using the equation below and what we've been using all lesson:
In the following problems, students will continue to work with the formula for electrostatic potential energy by using it to solve for the distance between two spheres given the electrostatic potential energy and the charges of the spheres, as well as using the formula to solve for the charge of one of the spheres given the electrostatic potential energy, the distance between the spheres and the charge of the other sphere. After completing the beginning problems, students should move on to the challenge problems, which are more theoretical.
Find the distance between two spheres of charges, 0.0003 C and -0.0001 C, respectively, if the electrostatic potential energy is -450 J.
If the electrostatic potential energy between two spheres a distance of 2 meters apart is 100000 J, find the charge of the second sphere given that the first sphere has a charge of -0.005 C.
Both problems use the formula:
We have -450 = (9 x 10^9) * ((0.0003) * (-0.0001)) / r. Solving for r, we have r = -270 / -450 = 0.6 meters.
We have 100000 = (9 x 10^9) * ((-0.005) * q) / 2. Solving for the other charge q, we have q = 200000 / -45000000 = -0.0044 C.
What happens to the electrostatic potential energy between two spheres at a fixed distance apart if the charges of both spheres are doubled?
If one charge is tripled and the other charge remains the same, what would have to happen to the distance between the spheres to keep the electrostatic potential energy the same?
1. Using the formula:
Replace each charge with the charge doubled:
This shows us that the electrostatic potential energy is quadrupled.
2. Using the formula and replacing one charge with three times the charge and replacing the distance r with x, we have:
Since we want the electrostatic potential energy to stay the same, we set these two equations equal to each other and solve for the new distance x.
So, the spheres will have to be three times the distance apart to keep the electrostatic potential energy the same.
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