Coulomb's Law/Electric Fields I
Chapter 3: Coulomb’s Law/Electric Fields I
Chapter Learning Objectives: After completing this chapter the student will be able to: Use Coulomb’s Law to determine the force exerted by one electric charge on another. Determine the electric field surrounding a point charge. Determine the force on a charge due to an electric field. Use Superposition to determine the electric field due to multiple charges.
You can watch the video associated with this chapter at the following link:
Historical Perspective: Charles-Augustin de Coulomb performed pioneering work in electrostatics (and friction) in France in the late 1700s and early 1800s. Not only does he have a law named after him, but the fundamental unit of charge is also named the Coulomb.
Photo credit: https://commons.wikimedia.org/wiki/File:Charles_de_coulomb.jpg, [Public domain], via Wikimedia Commons.
3.1 Electric Charge
“Charge” is a characteristic of matter associated with protons (which possess positive charge) and electrons (which possess negative charge). Nearly every phenomenon we will study in this book arises from the interactions of electrical charges.
The fundamental SI unit for charge is the Coulomb, or C. A coulomb is a very large amount of charge! Considering that an electron and a proton both possess 1.602 x 10-19 C of charge, this means that one coulomb of charge contains 6.242 x 1018 individual electrons or protons. That is 6,242,000,000,000,000,000 or over 6 quintillion individual charges. A lighting strike, one of the most powerful and destructive naturally occurring electrical events, contains only about 10 to 20 coulombs of charge.
3.2 Electrical Forces and Coulomb’s Law
There are only a handful of universal rules in this field, but one of them is this:
Universal Rule #1: Electric charges always exert a force on each other.
If the two charges have the same polarity (both positive or both negative), this force will repel them from each other. If the charges have opposite polarity (one positive and one negative), then the force will cause them to be attracted toward each other.
The magnitude of the force is proportional to the magnitude of each charge and inversely proportional to the square of the distance between them, as shown in Equation 3.1:
This equation, which is one of the most fundamental in all of electromagnetic fields, is named
Coulomb’s Law. The constant 0 is referred to as the “permittivity of free space,” and its value is:
You might wonder why the 4 is separated out rather than being combined into the new constant. I promise we’ll explain that in just a couple of chapters.
The unit vector indicating the direction of the force is in the direction of the vector pointing from one charge to the other:
Here, r1 refers to the location of the charge that is “causing” the force, and r2 is the location of the charge that is “feeling” the force.
This equation and the constant 0 are set up so that if the charges are in units of Coulombs, and if the distance between the points is in meters, then the force will be in Newtons.
Example 3.1: What is the force exerted by charge B on charge A in the following figure? y (meters)
A (2C) 5 B (-3C) 3
x (meters) 2 4
3.3 Electric Fields
Coulomb’s Law can be a little bit difficult to work with, because there are four different quantities to keep track of—two charges and two locations. What would be more useful would be an equation or a quantity where we had some of the variables fixed and could investigate variations among the others. The electric field is a theoretical construct that allows us to do just this.
In calculating the electric field, we temporarily remove the second charge (the one that “feels” the force), and we ask ourselves, “What would be the force on a theoretical charge if it were placed at every possible location around the first charge?” This is the definition of the electric field of the first charge, and it exists at all points in space.
We can define the electric field caused by a fixed charge Q as being the force that would exist on a “test charge” q placed in the neighborhood of Q. By convention, we use a positive test charge. The electric field is defined to be the force on a positive test charge divided by the magnitude of the test charge.
Combining this definition of the electric field with Coulomb’s Law (Equation 3.1), we get the electric field surrounding a point charge:
The units of the electric field can be written as N/C or as V/m. We will find circumstances where each of these choices will be preferable.
Because the charge Q is not moving, we refer to this as an “electrostatic” field. The next seven chapters will focus on different aspects of electrostatics. (Things get more complicated when the charges start to move.)
Remember that we decided to use a positive test charge. This means that if the fixed charge is also positive, then the force will repel the test charge, which means the electric field will point away from a positive charge. Similarly, if the fixed charge is negative, it will attract a positive
4 test charge, which means that the electric fields will point toward a negative charge. This is shown in Figure 3.1.
Figure 3.1. Electric field lines point away from positive charges and toward negative charges.
If we want to be very fancy, we would say that electric fields “emanate” away from a positive charge and “terminate” on a negative charge. We could (and do) also say that positive charges are a “source” of electric fields, while negative charges are a “sink” of electric fields.
Example 3.2: A fixed charge of +3C is located at (2,0). What is the function E(x,y) for the electric field at all points in the (x,y) plane?
Example 3.3: Given the situation in Example 3.2, what is the electric field at the origin?
Example 3.4: Given the situation in Examples 3.2 and 3.3, what would be the force on a charge of -4C located at the origin?
3.4 Superposition of Electric Fields
Electric fields obey superposition, which means that if there are multiple charges near a point, you can calculate the total electric field at a given point by calculating the electric fields due to each charge and then adding them together. Of course, electric fields are vector quantities, so you must use vector addition. Superposition of electric fields is defined mathematically in equation 3.6:
Example 3.5: A charge of +4mC is located at (-3,0), and a charge of -3mC is located at (+4,0). Use superposition to determine the electric field at the origin. What is the force on a charge of +2C at the origin?
Example 3.6: Three charges, each of +1C, are located at three corners of a square: (-3,-3), (-3, +3), and (+3, -3). What is the electric field at the fourth corner (+3, +3)? What is the force on a charge of +1nC at (+3,+3)?
Example 3.7: Superposition can also be analyzed using sketches of electric field lines, which always begin on positive charges and end on negative charges. An “electric dipole” includes one positive and one negative charge, typically of equal magnitude. Sketch the electric field lines associated with the following electric dipole:
Universal Rule #1: Electric charges always exert a force on each other. Coulomb’s Law specifies the magnitude and direction of the force caused by one electric charge on another electric charge. The electric field removes one of the two charges and answers the question, “What would be the force on a conceptual positive charge caused by the fixed charge that remains?” Electric fields point away from positive charges and toward negative charges. Electric fields obey superposition: the electric field at a given point can be calculated as the sum of the electric fields due to all surrounding charges.