Electromagnetism

Electric Field

Here

Define the electric field as force per unit charge, build it from point charges, and explore superposition and field lines.

Electric Potential

Next

Move from force to energy: potential energy landscapes, the electric potential, and its relationship to the field.

Electric Current

Finish

Follow charges into motion: the Drude model of electron drift, Ohm’s law, and series–parallel circuits.

Electric Field

An electric field is the force per unit charge at a point in space. We define it by

E⃗=F⃗qtest,\vec{\mathbf E} = \frac{\vec{\mathbf F}}{q_{\text{test}}},

where F⃗\vec{\mathbf F} is the electric force acting on a small, positive test charge qtestq_{\text{test}}. In other words, the field describes the force a test charge would feel if placed at a particular location.

If vector components and unit vectors feel rusty, the Vectors page is a good warm-up — the field is a vector at every point in space.

Point Charge

In the presence of a single point charge, the electric field is

E⃗=k qr2 r^,\vec{\mathbf E} = k\,\frac{q}{r^2}\,\hat{\mathbf r},

where kk is Coulomb’s constant and qq is the charge in coulombs.

Do these two definitions of the electric field agree with Coulomb's law?

Start from Coulomb’s law for the force on a test charge in the presence of a point charge:

F⃗=k q qtestr2 r^.\vec{\mathbf F} = k\,\frac{q\,q_{\text{test}}}{r^2}\,\hat{\mathbf r}.

Divide both sides by the test charge to get the electric field:

E⃗=F⃗qtest=k qr2 r^.\vec{\mathbf E} = \frac{\vec{\mathbf F}}{q_{\text{test}}} = k\,\frac{q}{r^2}\,\hat{\mathbf r}.

The vector r⃗\vec{\mathbf r} points from the charge’s position to the location of interest where we are computing the field, so its magnitude r=∥r⃗∥r = \lVert \vec{\mathbf r} \rVert is the distance from the charge. The direction of E⃗\vec{\mathbf E} comes from the unit vector r^=r⃗∥r⃗∥\hat{\mathbf r} = \dfrac{\vec{\mathbf r}}{\lVert \vec{\mathbf r} \rVert}.

Try it:

  • Double qq and confirm E∝qE \propto q.
  • Halve rr and confirm E∝1/r2E \propto 1/r^2.
  • Flip the sign of qq and watch the field direction reverse.

Superposition

When multiple charges are present, the net electric field is the vector sum of the individual fields:

E⃗net=∑iE⃗i.\vec{\mathbf E}_{\text{net}} = \sum_i \vec{\mathbf E}_i.

With two point charges, the net field is simply E⃗net=E⃗1+E⃗2\vec{\mathbf E}_{\text{net}} = \vec{\mathbf E}_1 + \vec{\mathbf E}_2, added tip-to-tail.

Try it:

  • Place two equal charges and find the point where E⃗=0\vec{\mathbf E} = 0.
  • Make a dipole (+q+q, −q-q) and describe the shape of the net field.

Is there one electric field, or many?

We often speak of individual electric fields due to each point charge. This is a useful mental model for connecting to Coulomb’s law and applying superposition.

In the long run, it is better to imagine that space is filled with a single, unified field E⃗net\vec{\mathbf E}_{\text{net}}. The field is a property of space itself, shaped by the configuration of charges. The field then determines the force on charges within that space, which predicts their motion, which changes the field, and so on:

charge configurationelectric fieldforces on chargesmotion of charges

Electric Force and Field Lines

The whole point of the field is that, once we know it, the net force on any charge placed in it is

F⃗net=qtest E⃗net.\vec{\mathbf F}_{\text{net}} = q_{\text{test}}\,\vec{\mathbf E}_{\text{net}}.

The explorer below draws the field around a configuration of charges. Use the presets for a monopole, dipole, or capacitor; switch on field lines; and launch a green test charge to watch it accelerate along the force qE⃗q\vec{\mathbf E}.

Electric Field Explorer

Place charges, choose a preset, draw field lines, and launch a test charge.

Did we add any new physics?

Not yet. So far the electric field is just a way to organize our thinking about forces between charges. Notice that the field is implicitly a function of position (and time), since the force on a test charge depends on where it sits:

E⃗=E⃗(r⃗,t).\vec{\mathbf E} = \vec{\mathbf E}(\vec{\mathbf r}, t).

In these simulations the field updates instantaneously as charges move. That is not how reality works — the field concept becomes far more powerful in genuinely time-dependent situations, where changing electric fields generate magnetic fields and vice versa. That is the road to electromagnetism.

Problem Solving

Field of a Point Charge

Problem

Find the magnitude of the electric field 0.5 m0.5\,\text{m} from a 2 μC2\,\mu\text{C} point charge.

Given

  • k≈9×109 N⋅m2/C2k \approx 9 \times 10^9\,\text{N·m}^2/\text{C}^2
  • q=2×10−6 Cq = 2 \times 10^{-6}\,\text{C}
  • r=0.5 mr = 0.5\,\text{m}

Solve

E=k ∣q∣r2=9×109⋅2×10−6(0.5)2≈7.2×104 N/C.E = k\,\frac{|q|}{r^2} = 9 \times 10^9 \cdot \frac{2 \times 10^{-6}}{(0.5)^2} \approx 7.2 \times 10^4\,\text{N/C}.

Net Field at a Midpoint

Problem

Two identical 1 μC1\,\mu\text{C} point charges are placed 1 m1\,\text{m} apart. Determine the net electric field at the midpoint between them.

Solve

At the midpoint, each charge contributes a field of equal magnitude pointing in the opposite direction (both fields point away from their like-signed source). The two contributions cancel:

E⃗net=E⃗1+E⃗2=0.\vec{\mathbf E}_{\text{net}} = \vec{\mathbf E}_1 + \vec{\mathbf E}_2 = 0.

Why E = 0 Inside a Conductor

Problem

In electrostatic equilibrium (no motion of free charges), why must the electric field be zero inside a conductor?

Reasoning

If the field were nonzero, the free charges inside would feel a force F⃗=qE⃗\vec{\mathbf F} = q\vec{\mathbf E}, and a net force means acceleration — not equilibrium. The charges keep rearranging until their own field exactly cancels the interior field everywhere, at which point the motion stops. That equilibrium configuration is precisely the one with E⃗=0\vec{\mathbf E} = 0 inside.

Electric Field Checkpoint

Question 1 of 3

What does an electric field tell you at a point in space?

Choose an answer to get instant feedback.