Electric Field

Core Idea

The electric field E at a point in space is defined as the force per unit positive test charge placed at that point: E = F/q. It is a vector field, meaning it has both magnitude and direction at every location. For a point charge Q, E = kQ/r² directed radially outward (for positive Q). The superposition principle allows the total field from multiple charges to be found by vector addition of individual contributions.

How It's Best Learned

Draw field vectors at several points around simple charge distributions before moving to field-line diagrams. Work problems computing E at specific points before tackling more abstract questions about field patterns.

Common Misconceptions

• The electric field exists independently of any test charge — it is a property of space created by source charges.
• Field lines never cross; if they did, the field would have two directions at one point.
• A uniform field (like between parallel plates) does not mean the charges are at rest.

Explainer

From Coulomb's law you know that two charges exert forces on each other: F = kQq/r². But this framing has a subtle problem — it suggests that charges act on each other directly across empty space, which puzzled physicists for centuries. The electric field concept reframes this: instead of saying charge Q pushes on charge q, we say charge Q *creates a field throughout space*, and that field then acts locally on q. The field is real and exists whether or not there is anything to feel it.

Formally, the electric field E at a point is defined as the force a positive test charge would experience *per unit charge* if placed there: E = F/q. The test charge is conceptual — a hypothetical +1 C probe. If a real +2 μC charge at a point experiences 8 N to the right, the field there is 8/2×10⁻⁶ = 4×10⁶ N/C to the right. Crucially, that field value does not depend on the test charge you use to measure it. Double the test charge, the force doubles, but E = F/q stays constant. The field is a property of the location, created by the source charges.

For a single point charge Q, the field at distance r is E = kQ/r², directed radially outward if Q is positive and inward if Q is negative. Notice the structural similarity to Coulomb's law — E just replaces F/q, with q factored out. This means you already know how E falls off with distance (inverse-square) and how it depends on the source charge.

When multiple source charges are present, the superposition principle lets you find the total field by vector addition. Each source charge contributes its own field independently; you add the vectors. This is why the field at the exact midpoint between two equal positive charges is zero — the rightward field from the left charge and the leftward field from the right charge cancel perfectly. Always treat this as vector addition, not scalar addition, or you will get wrong answers whenever contributions point in different directions.

Field lines are a visualization tool for understanding field patterns. They point in the direction a positive test charge would move, and their density indicates field strength. Two rules follow from the field's mathematical properties: field lines start on positive charges and end on negative charges, and they never cross (because the field has exactly one direction at every point). These patterns will become critical when you reach Gauss's law and electric potential, where the geometry of field lines carries quantitative information.