Electric Fields
Instead of thinking about the force between two charges directly, we split it into two steps:
- A charge creates a field everywhere in space.
- Another charge placed in the field feels a force.
The electric field $\mathbf{E}$ at a point is the force per unit positive charge:
$$\mathbf{E} = \frac{\mathbf{F}}{q}$$
If you placed a tiny positive test charge at that point, $\mathbf{E}$ tells you which way it would be pushed, and how hard.
Field from a point charge
A single charge $Q$ creates a radial field:
$$\mathbf{E} = k \frac{Q}{r^2} \hat{\mathbf{r}}$$
- Positive charge: field points outward (it would push a positive test charge away)
- Negative charge: field points inward (it would pull a positive test charge in)
- Strength falls off as $1/r^2$ — twice as far away, one quarter the strength
Superposition
The field from multiple charges is just the vector sum of their individual fields. This superposition principle means we can build up complex fields from simple pieces.
Click to place charges. Each charge creates a field; the total field at every point is the vector sum. Red = positive, blue = negative.
Field lines
A useful visualization: draw curves that follow the direction of $\mathbf{E}$ everywhere. These field lines:
- Start on positive charges and end on negative charges (or go to infinity)
- Never cross each other (the field has one direction at each point)
- Are denser where the field is stronger
The interactive above shows the field as arrows. Field lines give a complementary view — they emphasize the global flow pattern rather than the local value at each point.
Why the field concept matters
The field idea seems like an unnecessary abstraction — why not just compute forces directly? Three reasons:
- Fields propagate at the speed of light. If a charge moves, the change in force doesn’t reach a distant charge instantly. The field carries the information.
- Fields carry energy. The energy of the electromagnetic field is real and measurable.
- Fields are the natural language of Maxwell’s equations. All four equations describe how the electric and magnetic fields behave — where they diverge, where they curl, and how they change in time.