Electric Field You have learned that two charges will exert a force on each other even though they are not actually touching each other. This force is called the Coulomb force. For two charges q 1 and q 2 separated by a distance R this force law takes the form: A natural question to ask would be “How does one charge know that the other charge is present?” What actually happens is that every charge creates an electric field in the space surrounding it. Other charges then do not interact directly with the charge. Rather they interact with the field. In the diagram above, q 1 creates an electric field all around it. The charge q 2 experiences this field at the point a distance R away from q 1. Of course q 1 experiences the field created by q 2 as well.

To define the electric field precisely, we need to introduce the concept of a “test charge.” A test charge is defined to be a small positive charge. The electric field at some point in space is defined as the force exerted on a test charge placed at that point, then divided by the numerical value of the test charge. The figures below show how the field is defined: charges creating field Point in space where you want to know field Place test charge at the point and measure force on it Electric field points in direction of force on test charge Definition of Electric Field

To actually measure the electric field created by some charges at a point, you have to place a test charge at the point and then measure the force on the test charge. Then you divide the force by the numerical value of the test charge. It is important to recognize that the electric field at some point is created by all the other charges in the region. Putting another charge at that point does not change the field at that point. Also, the electric field is a vector quantity: it has a direction. The electric field at a point always points in the direction in which a positive charge would move if placed at that point. This means that negative charges will always feel a force in the opposite direction of the field. Example: A +4 µC is placed in a region of space where there are several charges. It experiences a force of 0.05 N in the + x direction. Calculate the value of the electric field and determine what direction it is in. Use the formula from the definition to calculate the value of the field. The direction of the field is in the + x direction, since this is the direction a + charge is forced.

Once you know the value of the electric field at a point, you can use this value to calculate the force on any charge placed at that point. This is because the value of the field does not depend on the actual charge placed at the point, only all the other charges that are creating the field at that point. Example: Suppose a -10 µC charge is placed at the same point as in the previous example. Calculate the force it would experience and determine the direction of this force. You can rearrange the equation for the electric field to get: Since we already know E, we can substitute Note that you do not need to put the sign of the charge in since you just calculating the size or magnitude of the force. The direction of the force is opposite the field since it is negative charge placed in the field. Thus the direction is in the –x direction.

Electric Field due to a Single Charge It is possible to use the definition of the electric field to derive a formula for the electric field that a single charge creates around it. Suppose you have a charge Q and you want to calculate the field created by this charge a distance R away from it. Want to calculate field at this point Imagine placing a test charge q at this point. It will experience a force as shown below. Then use the definition of the electric field to get a formula for the field at a distance R away from the charge Q: Field created by a single charge

Exercise: A + 5 µC charge is located at the origin. Calculate the magnitude of the electric field created by this charge at the point (-4, 0) and determine the direction of the field. The point at which you want to calculate the field is 4 m from the charge creating the field. Use the formula just derived to calculate the size of the field. To determine the direction of the field, you ask yourself what direction a positive charge (test charge) would move if placed at that point. Since positive charges repel, the test charge would move to the left. This means the field points to the left, in the –x direction.

Electric Field Created by More Than One Charge Suppose you have several charges in a given region. Each of the charges creates its own electric field in the space around it. How can you determine what the overall or net field is at some point? Use the Principle of Superposition: to find the net electric field, calculate the individual field created by each charge as if no other charges were present. The total electric field or the net electric field is the vector sum of each of the individual fields created by each charge separately. Since you know how to calculate the field of a single charge, the above prescription may seem pretty straightforward. However, because the electric field is a vector, you cannot simple add the size of each separate field vector; you need to take the direction into account as well. To simplify things, we will work with charges that lie along a single line. Then taking directions into account will mean determining whether to add or subtract the sizes of each electric field vector. An example will demonstrate how this works.

Example: A -4 µC charge is located at the point (-2, 0) and a +2 µC charge is located at the point (1, 0). Calculate the size of the electric field at the point ( 3, 0) and also determine the direction of the field at this point. Want to calculate field at this point The Principle of Superposition tells you to find the field created by each of the charges separately. First draw in the vector representing the field created by each charge at the point. Note that you determine the direction of each separate field by asking what would happen to a + test charge placed at the point. Now calculate the size of each separate field. Since you just need the size, do not bother with the sign of the charge. From the diagram, you see that the two contributing fields point in the opposite direction. This means that you must subtract them when you combine them to find the net field. Since E + is larger, the net field will point in the +x direction.

Electric Field Lines The electric field is not a tangible property: you cannot see it or touch it. However, it is possible to represent the field graphically so that you can better visualize it. This graphical representation of the electric field is called electric field lines. If you can draw the field lines in a given region, then you will be able to predict the effects of the electric field in that region on any charges that might be placed in that region. There are two basic rules for drawing electric field lines: 1. The electric field vector at any point is tangent to the field line drawn through that point 2. The strength of the electric field at a point is proportional to the density of the field lines at that point. Near a positive charge, the field lines point radially outward. Near a negative charge, the field lines point radially inward. The diagrams below illustrate this.

When several charges are present in a region, the field lines take on a more complex configuration. Note that the Electric field vectors drawn are tangential to the field lines. Below is a more complex example.

Motion of a Charge in an Electric Field When a charge enters a region of electric field, it feels a force. From Newton’s 2 nd law, you know that a net force will produce an acceleration. This if a charge feels a net force caused by the electric field, you can combine the two ideas to predict the acceleration of a charge place in an electric field. Example: An electron enters a region where there is an electric field of magnitude 500 N/C directed along the –y axis. Calculate the acceleration of the electron and determine the direction of this acceleration. You need to look up the mass and charge of the electron and substitute into the equation. Since the field points in the –y direction, the force on the electron and therefore the acceleration of the electron will be in the +y direction, since the electron is negative. Why is the acceleration of the electron so large?

More Electric Field problems Two charges are placed on the x axis. The first is 1.5  C at (0,0) and the second is 6.0  C at (0.6,0). Where on the x axis is E = 0? A proton is traveling horizontally to the right with a speed of 2.0E5 m/s. (a) What E does it take to stop the proton in 4.5 cm? (b) How long (time) will the proton take to stop?