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Chapter 1: The Electric Field

The fundamental problem of electrostatics is to determine the force acting on a specific charge due to a configuration of other stationary charges. This requires a precise understanding of how charges interact across space.

0. What Is Charge?

Electric charge is a fundamental property of matter, written \(q\) and measured in coulombs (C). It tells us how matter participates in electric and magnetic interactions. In this lesson, charge is the thing that creates electric forces, and also the thing that feels electric forces.

Charge comes in two signs: positive and negative. Charges with the same sign repel each other, while charges with opposite signs attract. This sign rule is the physical starting point behind Coulomb's law.

Experimental facts about charge

  • Electric charge is a scalar property: its value does not change when we describe the same object from a different inertial reference frame.
  • In an isolated system, the total electric charge is conserved.
  • Isolated observable charges come in integer multiples of the elementary charge \(e = 1.602 \times 10^{-19}\,\text{C}\).
  • Charge is an independent property of a particle, not something determined by its mass.

Inside atoms

Protons carry positive charge, electrons carry negative charge, and neutrons carry no net charge.

Neutral objects

An atom or object is neutral when its total positive charge and total negative charge cancel.

Charged objects

An object becomes charged when there is an imbalance. In ordinary charging, electrons move while protons stay locked in nuclei.

We often say that the whole atom or whole object is charged because from outside the object, what matters is the net result after all positive and negative charges are added together:

$$ q_{\text{net}} = (N_p - N_e)e $$
\(N_p\) Number of protons.
\(N_e\) Number of electrons.
\(e\) The elementary charge, \(e = 1.602 \times 10^{-19}\,\text{C}\).

If \(N_p = N_e\), then \(q_{\text{net}} = 0\). If one electron leaves, then \(N_p > N_e\), so the object has positive net charge. If an extra electron is added, then \(N_e > N_p\), so the object has negative net charge.

We use the single symbol \(q_{\text{net}}\), or simply \(q\), because outside the atom or object the detailed list of every proton and electron is usually unnecessary. Replacing many internal charges by one net charge gives the same outside effect in many calculations, while making the problem much easier.

This is the first optional deep dive in the course. Deep dives are for slower explanations, examples, proofs, and extra context. The main lesson gives the ideas you need first; the deep dive lets you look under the hood when you want more.

Deep Dive: Atomic Picture of Charge

1. The Interaction of Two Charges

Consider a single Source Charge (\(q\)) situated at position \(\vec{r}'\) and a Test Charge (\(Q\)) situated at position \(\vec{r}\).

The force exerted on the test charge is defined by Coulomb's Law:

$$ \vec{F}_Q = \frac{1}{4\pi\epsilon_0} \frac{q Q}{\rsep^2} \hatrsep $$
\(q\) The Source Charge. The charge creating the force.
\(Q\) The Test Charge. The charge experiencing the force.
\(\vec{r}'\) The source position vector. It locates the source charge \(q\).
\(\vec{r}\) The test position vector. It locates the test charge \(Q\), where the force is evaluated.
\(\epsilon_0\) Permittivity of Free Space. A physical constant \(\approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2\).
\(\vecrsep\) The Separation Vector \(\vecrsep = \vec{r} - \vec{r}'\). It points from the source charge to the test charge.
\(\rsep\) The magnitude (distance) between the charges: \(\rsep = |\vecrsep|\).
\(\hatrsep\) The unit vector defined as \(\hatrsep=\frac{\vecrsep}{\rsep}\). This specifies the direction of the force purely.

This formula says that the force depends on three ideas at once: the two charge amounts \(q\) and \(Q\), the distance \(\rsep\) between them, and the direction from the source charge to the test charge \(\hatrsep\). Bigger charges make a bigger force, while larger separation weakens the force by the inverse-square rule \(1/\rsep^2\). The constant \(\epsilon_0\) is the permittivity of free space, meaning vacuum. Later, when charges interact inside materials instead of vacuum, \(\epsilon_0\) is replaced by the material permittivity \(\epsilon\).

You can see each part of the formula in the simulator below: the charges \(q\) and \(Q\), the separation vector \(\vecrsep\), its direction \(\hatrsep\), and the resulting force on each charge.

Deep Dive: Coulomb's Law

2. The Principle of Superposition

In most physical situations, there is more than one source charge. Experiments confirm that the electromagnetic force obeys the Principle of Superposition.

The total force on a test charge \(Q\) is the vector sum of the individual forces exerted by each source charge \(q_i\):

$$ \vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \dots + \vec{F}_n = \sum_{i=1}^n \vec{F}_i $$

3. The Electric Field

If we examine the superposition equation, we observe that the test charge \(Q\) is a common factor in every term. We can factor it out:

$$ \vec{F} = Q \left[ \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{\rsep_i^2} \hatrsep_i \right] $$

The quantity inside the brackets depends only on the distribution of the source charges and the position in space. To measure it without significantly disturbing the source charges, we imagine a very small positive test charge \(Q\). The resulting quantity is independent of the particular test charge used. We define this vector field as the Electric Field, \(\vec{E}\).

$$ \vec{E}(\vec{r}) \equiv \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{\rsep_i^2} \hatrsep_i $$

Thus, the force on any charge \(Q\) placed in an external electric field is simply \(\vec{F} = Q\vec{E}\).

In the simulator below, choose a scenario to see how the source charges shape the electric field. Field lines show the direction a positive test charge would be pushed, while vector samples show the local direction and relative strength of \(\vec{E}\).

Deep Dive: Electric Field Visualizations

4. Continuous Charge Distributions

Macroscopic objects contain such a vast number of charge carriers that it is impractical to perform a summation. Instead, we treat the charge as being spread continuously over the object.

From Summation to Integration

We divide the charged object into infinitesimal elements \(dq\). The sum becomes an integral. To perform this integration, we must express \(dq\) in terms of geometry using Charge Density (line charge \(\lambda\), surface charge \(\sigma\), or volume charge \(\rho\)).

Deep Dive: Continuous Charge

5. The General Integral Formulation

Combining the concept of the electric field with continuous charge densities, the Electric Field at a position \(\vec{r}\) generated by a continuous volume of charge is:

$$ \vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int_V \frac{\rho(\vec{r}')}{\rsep^2} \hatrsep \, d\tau' $$

Note on vectors in integrals: The unit vector \(\hatrsep\) depends on the position of the source element \(dq\). As you integrate over the volume, \(\hatrsep\) changes direction. Therefore, it cannot be treated as a constant and must remain inside the integral.

Deep Dive: Calculating Fields: The Integral Method