All particles have certain intrinsic properties. The most commonly known property of a particle (or collection of particles) is the mass. As we have studied in classical mechanics, a system of masses interacts through the gravitational force, as Newton studied back in 1687 in his Principia Mathematica. Another property of all matter is the electric charge. Just as the mass of an object is intimately familiar (we all have to lift objects every day!), so is electric charge. When shuffling across a fluffy carpet and then touching a metal door handle, one experiences an electric shock, as electric charge is transferred between your hand and the door handle. In contrast to mass, which is always positive, electric charge can either be positive or negative. For example, when you put a battery into a remote control, the battery has a positive and negative side. Positive charge comes out of the \(+\) side, and goes in to the negative side. The battery must be plugged in the right way, otherwise it does not work!

To get an understanding of these electric charges, we first need to study how they interact with each other. In the same way that Newton’s law of gravitation tells us how strongly two masses attract one another, there is an analogous law for electric charges, known as Coulomb’s law. Coulomb’s law is the statement that the force of \(q_a\) on \(q_b\) is given by

\[\begin{equation} \mathbf{F}_{ab} = \frac{q_aq_b}{4\pi\epsilon_0 r^2}\hat{r}_{ab}, \end{equation}\]

where \(\hat{r}_{ab}\) is the unit vector which points from \(q_a\) to \(q_b\). The constant \(\epsilon_0 = 8.85\times 10^{-12}\text{C}^2/\text{N}\text{m}^2\) is called the permittivity of free space. Coulomb’s law is known to very high precision. It goes as \(1/r^n\), where \(n\) has been measured experimentally to be very close to 2, in fact \( |n-2| < 10^{-16}\).

Function

Schematic of Coulomb's law for the case where \(q_a\) and \(q_b\) are like charges. Because the two charges are like, they repel one another.

From Coulomb’s law, we observe that:

  • If \(q_a\) and \(q_b\) are both positive, or both negative, the force is repulsive
  • If \(q_a\) and \(q_b\) are opposite in sign, then the force is attractive

Surrounding a point charge is the electric field, and the electric field due to an isolated charge \(Q\) is given by

\[\begin{equation} \mathbf{E} = \frac{Q}{4\pi\epsilon_0 r^2}\hat{r}. \end{equation}\]

Function

Schematic of the electric field surrounding a point charge \(Q\). Note that the field lines are not drawn to scale. In reality, they should decay much faster and therefore the ones far from the charge should be much shorter. Left: This is for the case \(Q > 0\), where the field lines point away from the charge. Right: This is for the case \(Q < 0\), where the field lines point toward the charge.

Regardless of what produces the electric field, we can find the force due to this electric field \(\mathbf{E}\) on some charge \(q\) placed in it. The force on a test charge \(q\) is given by \(\mathbf{F} = q\mathbf{E}\). Before moving on to some examples, let’s note that, in these notes, we will be using SI units. Important values in these units are

  • The mass of the electron \(m_e = 9.1\times 10^{-31}\text{kg}\)
  • The mass of the proton: \(m_p = 1826m_e\)
  • The charge of the proton: \(e = 1.6\times10^{-19}\text{C}\). This is also called the elementary charge.
  • The speed of light: \(c = 2.9979\times10^8\text{m}/\text{s}\).

A further important fact, which may seem trivial, is the law of superposition of electric fields. This states that, the electric field due to multiple charges is the sum of the electric fields of each individual charge. To see this, we can use the fact that forces can be added together. Suppose we have \(n\) point charges \(q_1,q_2,\dots,q_n\). The force on a charge \(q\) is given by

\[\begin{equation} \mathbf{F} = \sum_{i=1}^n \frac{qq_i}{4\pi\epsilon_0 r_i^2}\hat{r}_i \end{equation}\]

Factoring out \(q\), we use the relation between force on a charge \(q\) and the electric field it sites in: \(\mathbf{F} = q\mathbf{E}\). This allows us to conclude that

\[\begin{equation} \mathbf{E} = \sum_{i=1}^n \frac{q_i}{4\pi\epsilon_0 r_i^2}\hat{r}_i. \end{equation}\]

So we have derived superposition of electric fields from superposition of forces.

Example: Let us use the fact of superposition of electric fields to do the following problem. Suppose we have two point charges \(Q_1\) and \(Q_2\) at \(x = - a\) and \(x = a\), respectively. Calculate the electric field at \((x,y)\) in space.

Function

Solution: The magnitude of the electric field due to charge 1 is given by $$\begin{equation} E_1 = \frac{Q_1}{4\pi\epsilon_0 r_1^2} = \frac{Q_1}{4\pi\epsilon_0[(a+x)^2 + y^2]} \end{equation}$$ and, now putting in the unit vector which points from \((-a,0)\) to \((x,a)\) $$\begin{equation} \mathbf{E}_1 = E_1(\cos\theta_1\xhat + \sin\theta_1\yhat) = E_1\left(\frac{a+x}{r_1}\xhat + \frac{y}{r_1}\yhat\right) \end{equation}$$ where \(r_1 = \sqrt{(x+a)^2 + y^2}\), and similarly, $$\begin{equation} E_2 = \frac{Q_2}{4\pi\epsilon_0 r_2^2} = \frac{Q_2}{4\pi\epsilon_0[(x-a)^2 + y^2]} \end{equation}$$ and, adding the unit vector of the direction, we get $$\begin{equation} \mathbf{E}_2 = E_2(\cos\theta_2\xhat + \sin\theta_2\yhat) = E_2\left(\frac{x-a}{r_2}\xhat + \frac{y}{r_2}\yhat\right) \end{equation}$$ where \(r_2 = \sqrt{(x-a)^2 + y^2}\). By the principle of superposition, \(\mathbf{E} = \mathbf{E}_1 + \mathbf{E}_2\), and the final answer is, for the special case \(Q_1 = Q_2\) (also write for \(Q_1\neq Q_2\)) $$\begin{equation} \mathbf{E} = \frac{Q}{4\pi\epsilon_0}\left\{\left[\frac{x+a}{[(x+a)^2 + y^2]^{3/2}} + \frac{x-a}{[(x-a)^2+y^2]^{3/2}}\right]\hat{x} + \left[\frac{y}{[(x+a)^2+y^2]^{3/2}} + \frac{y}{[(x-a)^2+y^2]^{3/2}}\right]\hat{y}\right\} \end{equation}$$