Electric field lines

Electric field lines are a way of representing an electric field in a diagram. Their direction shows which way the field acts, and their spacing indicates the field’s relative strength.

Field direction and strength

At any point, the direction of a field line is the direction of the force that a positive test charge would experience there. Arrows on the lines show this direction: field lines point away from positive charges and towards negative charges. A negative test charge would experience a force in the opposite direction.

Field lines drawn closer together represent a stronger field; lines drawn farther apart represent a weaker field. Their spacing therefore allows the relative field strength at different points to be compared.

Field lines do not cross. If two lines crossed, they would indicate two different field directions at the same point, but the electric field has a single direction at each point.

Uniform fields and point charges

A uniform electric field has the same direction and strength throughout the region represented. It is shown by straight, parallel, equally spaced field lines: parallel lines show the same direction, while equal spacing shows the same strength.

The field lines of a point charge radiate in all directions. They point outwards from a positive point charge and inwards towards a negative point charge. The lines spread farther apart with distance from the charge, representing a field that becomes weaker farther away.

Electric field strength

An electric field exerts a force on a charge placed in it. Electric field strength at a point describes the force per unit positive charge at that point. A stronger field exerts a greater force on the same charge.

Here, $$E$$ is the magnitude of the electric field strength, $$F$$ is the magnitude of the electric force, and $$Q$$ is the magnitude of the charge. Electric field strength can be measured in newtons per coulomb, $$\mathrm{N,C^{-1}}$$, or equivalently in volts per metre, $$\mathrm{V,m^{-1}}$$.

Electric field strength is a vector: it has both magnitude and direction. Its direction is defined as the direction of the force on a positive test charge. A negative charge placed at the same point experiences a force in the opposite direction to the field.

Electric field strength between parallel plates

In a uniform electric field, the field strength is the same at every point. Between parallel plates, it depends on the potential difference across the plates and their separation in the field direction:

Here $$E$$ is the electric field strength, $$V$$ is the magnitude of the potential difference between the plates, and $$d$$ is their separation. Field strength can be expressed in volts per metre ($$\mathrm{V,m^{-1}}$$).

Electric field strength and energy change

Consider a charge of magnitude $$Q$$ moving between the plates through distance $$d$$ in the field direction. If the magnitude of the electric force is $$F$$, the magnitude of the work done by that force is $$Fd$$. This equals the magnitude of the electrical potential energy change, $$Q\Delta V$$, where $$\Delta V$$ is the magnitude of the potential difference crossed:

Since $$F = QE$$, substitution gives $$QEd = Q\Delta V$$. Cancelling $$Q$$ and dividing by $$d$$ gives $$E = \Delta V/d$$. For the full separation of the plates, $$\Delta V = V$$.

These equations use magnitudes: the signed work done by the electric force is opposite to the signed change in electrical potential energy. The field direction is towards decreasing electric potential.

Charged-particle motion in a uniform electric field

A charged particle entering a uniform electric field at right angles to the field continues to move across it while being accelerated along or against the field. Its path therefore bends while it remains in the field.

The electric force has constant magnitude:

Here, $$q$$ is the particle’s charge and $$E$$ is the electric field strength. The force is along the field for a positive charge and opposite to the field for a negative charge. Because the field is uniform, the force and hence the acceleration are constant, provided other forces are negligible.

Motion in two directions

The force acts only parallel to the field, so the particle has no acceleration perpendicular to the field. Its velocity component in that direction therefore remains constant. Its velocity component parallel to the field starts at zero and changes uniformly as the particle accelerates.

If the particle enters at speed $$v_0$$, take $$x$$ in its initial direction and $$y$$ along its acceleration. After time $$t$$, its displacements are $$x = v_0t$$ and $$y = \tfrac{1}{2}at^2$$, where $$a$$ is the acceleration magnitude. Eliminating $$t$$ gives:

The displacement along the acceleration is proportional to the square of the displacement across the field. The trajectory is therefore parabolic while the particle remains in the uniform field.

Electric field strength of a point charge

An isolated point charge produces an electric field that extends radially: at any point, the field lies along the straight line joining that point to the charge. Electric field strength describes the force per unit positive charge at that point, so it has both a magnitude and a direction.

In free space, the field-strength magnitude $$E$$ at a distance $$r$$ from a point charge $$Q$$ is:

Here, $$|Q|$$ is the magnitude of the source charge and $$\varepsilon_0$$ is the permittivity of free space. The magnitude $$E$$ is measured in newtons per coulomb ($$\mathrm{N,C^{-1}}$$).

Charge and distance

At a fixed distance, field strength is directly proportional to the magnitude of the source charge: doubling $$|Q|$$ doubles $$E$$.

Field strength also follows an inverse-square relationship with distance. At twice the distance, it is one-quarter as large; at three times the distance, it is one-ninth as large.

Direction of the field

The sign of the source charge determines the field direction:

• For a positive source charge, the field points radially outward, away from the charge.
• For a negative source charge, the field points radially inward, towards the charge.

The equation uses $$|Q|$$ to calculate magnitude; the sign of $$Q$$ must be considered separately to determine direction.