18.4 Electric Field: Concept of a Field Revisited

Electric Field

Electric field definition and strength

An electric field is the region around a charged object where another charged object experiences an electric force. You can't see the field directly, but you can observe its effects: it pushes or pulls on any charge placed within it.

The electric field is a vector quantity, meaning it has both magnitude and direction.

Field strength ($E$) is defined as the force per unit charge:

$E = \frac{F}{q}$

• $F$ is the electrostatic force on the test charge (in Newtons)
• $q$ is the test charge (in Coulombs)

For a point charge, the electric field strength at a distance $r$ from the source is found using Coulomb's law:

$E = \frac{kQ}{r^2}$

• $k$ is Coulomb's constant ($8.99 \times 10^9 \, \frac{N \cdot m^2}{C^2}$)
• $Q$ is the source charge creating the field
• $r$ is the distance from the source charge to the point where you're measuring the field

Notice that $E$ depends on the source charge and distance only. The test charge doesn't affect the field itself.

Direction matters:

• A positive source charge produces a field that points radially outward (away from the charge)
• A negative source charge produces a field that points radially inward (toward the charge)

When multiple charges are present, you find the total electric field at any point by adding the individual field vectors from each charge. This is the superposition principle. You need to add these as vectors (accounting for direction), not just as numbers.

Force calculation on test charges

Once you know the electric field at a location, finding the force on any charge placed there is straightforward:

$F = qE$

• $F$ is the force on the test charge (in Newtons)
• $q$ is the test charge (in Coulombs)
• $E$ is the electric field strength at the test charge's location (in N/C)

The direction of the force depends on the sign of the test charge:

• A positive test charge feels a force in the same direction as the electric field
• A negative test charge feels a force in the opposite direction of the electric field

Steps to find the force on a test charge:

1. Calculate the electric field strength at the location of the test charge. For a point source charge, use $E = \frac{kQ}{r^2}$.
2. Multiply the field strength by the magnitude of the test charge: $F = qE$.
3. Determine the direction: same as $E$ for a positive charge, opposite for a negative charge.

One more detail worth knowing: electric dipoles (like water molecules, which have a positive end and a negative end) experience a torque in a uniform electric field that tends to align them with the field. In a non-uniform field, they can also experience a net force that pulls them toward the region of stronger field.

Relationship of field strength to force

The relationship $F = qE$ tells you that force is directly proportional to both the field strength and the charge. If you double the electric field while keeping the charge constant, the force doubles. If you double the charge while keeping the field constant, the force also doubles.

The direction of the force depends on two things: the direction of the electric field and the sign of the charge.

• A positive charge (like a proton) accelerates in the same direction as the electric field
• A negative charge (like an electron) accelerates in the opposite direction of the electric field

This is why, for example, in a uniform electric field between two parallel plates, protons and electrons curve in opposite directions.

Field visualization and properties

Electric field lines are a tool for visualizing the field's direction and relative strength.

• Lines point in the direction of the field (away from positive charges, toward negative charges)
• Where lines are packed closely together, the field is stronger. Where they spread apart, the field is weaker.
• Field lines never cross each other, because the field can only point in one direction at any given point.

Equipotential surfaces are imaginary surfaces where the electric potential has the same value everywhere. These surfaces are always perpendicular to the electric field lines. No work is done when moving a charge along an equipotential surface.

Electric flux measures how much electric field passes through a given surface area. It's proportional to the number of field lines passing through that surface and becomes important when you study Gauss's law later.