5.2 Coulomb’s Law

Learning Objectives

After Chapter 5.2, you will be able to:

Calculate the electric field that a charge generates and the electric force between it and another charge
Recall the direction in which a negative or a positive electrostatic force will move two charges relative to one another
Relate distance and charge quantities to electrostatic force and electrostatic field magnitudes
Apply direction conventions to draw electric fields generated by a charged object:

Coulomb’s law quantifies the magnitude of the electrostatic force F_e between two charges:

where F_e is the magnitude of the electrostatic force, k is Coulomb’s constant, q₁ and q₂ are the magnitudes of the two charges, and r is the distance between the charges. Coulomb’s constant (also called the electrostatic constant) is a number that depends on the units used in the equation. In SI units, where ε₀ represents the permittivity of free space, The direction of the force may be obtained by remembering that unlike charges attract and like charges repel. The force always points along the line connecting the centers of the two charges.

A close examination of Coulomb’s law reveals that it is remarkably similar in form to the equation for gravitational force. In the electrostatic force equation, the force magnitude is proportional to the charge magnitudes, and this is similar to the proportional relationship between gravitational force and mass. In both equations, the force magnitude is inversely proportional to the square of the distance of separation. These similarities ought to help you remember both equations on Test Day.

Bridge

Notice how Coulomb’s law looks very similar to the gravitational force equation, but with a different constant and using charge rather than mass. It is this fact that should remind us that this equation is dealing with electrostatic force between two charges, just as the gravitation equation is dealing with the gravitational force between two bodies of mass. The gravitation equation is discussed in Chapter 1 of MCAT Physics and Math Review.

Example:

Negatively charged electrons are electrostatically attracted to positively charged protons. Because electrons and protons have mass, they will be gravitationally attracted to each other as well. What is the ratio of the electrostatic force to the gravitational force between an electron and proton?

Solution:

Both Coulomb’s law and the universal law of gravitation state that the attractive forces between the electron and proton vary as the inverse of the square of the distance between them. The ratio between these forces can be calculated by dividing their magnitudes:

Now the values can be plugged in:

Note that the electrostatic attraction between the electron and proton is stronger than the gravitational attraction by a factor of almost 10⁴⁰. Also, note that setting up all of the variables before working out the math simplifies the process because a number of the variables cancel out during the division.

Electric Field

Every electric charge sets up a surrounding electric field, just like every mass creates a gravitational field. Electric fields make their presence known by exerting forces on other charges that move into the space of the field. Whether the force exerted through the electric field is attractive or repulsive depends on whether the stationary test charge q (the charge placed in the electric field) and the stationary source charge Q (which actually creates the electric field) are opposite charges (attractive) or like charges (repulsive).

Key Concept

Electric fields are produced by source charges (Q). When a test charge (q) is placed in an electric field (E), it will experience an electrostatic force (F_e) equal to qE.

The magnitude of an electric field can be calculated in one of two ways, both of which can be seen in the definitional equation for the electric field:

where E is the electric field magnitude in newtons per coulomb, F_e is the magnitude of the force felt by the test charge q, k is the electrostatic constant, Q is the source charge magnitude, and r is the distance between the charges. The electric field is a vector quantity, and we will discuss the process of determining the direction of the electric field vector in a moment. Look closely: you can see that this equation for the electric field magnitude is derived simply by dividing both sides of Coulomb’s law by the test charge q. In doing so, we arrive at two different methods for calculating the magnitude of the electric field at a particular point in space. The first method is to place a test charge q at some point within the electric field, measure the force exerted on that test charge, and define the electric field at that point in space as the ratio of the force magnitude to test charge magnitude One of the disadvantages of this method of calculation is that a test charge must actually be present in order for a force to be generated and measured. Sometimes, however, no test charge is actually within the electric field, so we need another way to measure the magnitude of that field.

Key Concept

By dividing Coulomb’s law by the magnitude of the test charge, we arrive at two ways of determining the magnitude of the electric field at a point in space around the source charge.

The second method of calculating the electric field magnitude at a point in space does not require the presence of a test charge. We only need to know the magnitude of the source charge and the distance between the source charge and point in space at which we want to measure the electric field In this method, we need to know the value of the source charge to be able to calculate the electric field.

By convention, the direction of the electric field vector is given as the direction that a positive test charge would move in the presence of the source charge. If the source charge is positive, then the test charge would experience a repulsive force and would accelerate away from the positive source charge. On the other hand, if the source charge is negative, then the test charge would experience an attractive force and would accelerate toward the negative source charge. Therefore, positive charges have electric field vectors that radiate outward (that is, point away) from the charge, whereas negative charges have electric field vectors that radiate inward (point toward) the charge. These electric field vectors may be represented using field lines, as shown in Figure 5.2.

around positive: radially outward; around negative: radially inward — Figure 5.2. Field Lines around a Positive, a Negative, and a Neutral Source Charge

Field lines are imaginary lines that represent how a positive test charge would move in the presence of the source charge. The field lines are drawn in the direction of the actual electric field vectors and also indicate the relative strength of the electric field at a given point in the space of the field. When drawn on a sheet of paper, field lines look like the metal spokes of a bicycle wheel: the lines are closer together near the source charge and spread out at distances farther from the charge. Where the field lines are closer together, the field is stronger; where the lines are farther apart, the field is weaker. Because every charge exerts its own electric field, a collection of charges will exert a net electric field at a point in space that is equal to the vector sum of all the electric fields.

Key Concept

Field lines are used to represent the electric field vectors for a charge. They point away from a positive charge and point toward a negative charge. The denser the field lines, the stronger the electric field. Note that field lines of a single charge never cross each other.

Because electric field and electrostatic force are both vector quantities, it is important to remember the conventions for their direction. If the test charge within a field is positive, then the force will be in the same direction as the electric field vector of the source charge; if the test charge is negative, then the force will be in the direction opposite to the field vector of the source charge.