Define electric field strength and potential, apply Coulomb's law and the field of a point charge, and analyse the uniform field between parallel plates

What this dot point is asking

SEAB wants you to define electric field strength and electric potential, to apply Coulomb's law and the radial field of a point charge, and to analyse the uniform field between charged parallel plates. This mirrors the gravitational field thread, with the key difference that charges can be positive or negative, so forces can attract or repel.

The answer

Coulomb's law

Two point charges $Q_1$ and $Q_2$ a distance $r$ apart in a vacuum exert a force on each other:

where $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{F m}^{-1}$ is the permittivity of free space and $\dfrac{1}{4\pi\varepsilon_0} = 8.99 \times 10^9\ \text{N m}^2\text{C}^{-2}$. Like charges repel; unlike charges attract. The force is inverse-square, like gravity.

Electric field strength

The electric field strength at a point is the force per unit positive charge placed there:

with units $\text{N C}^{-1}$ (equivalently $\text{V m}^{-1}$). It is a vector, directed away from positive charge and toward negative charge. For a point charge $Q$:

Electric potential

The electric potential at a point is the work done per unit positive charge to bring it from infinity to that point:

Potential is a scalar. It is positive near a positive charge and negative near a negative charge, with zero taken at infinity. The potential energy of a charge $q$ is $E_p = qV$.

The uniform field between parallel plates

Two parallel plates with a potential difference $V$ across a separation $d$ produce a uniform field between them:

directed from the positive to the negative plate. The field lines are parallel and evenly spaced, so a charged particle between the plates feels a constant force $F = qE$. This is the basis of cathode-ray deflection and many particle experiments.

Examples in context

Example 1. Deflecting an electron beam. In a cathode-ray tube, electrons pass between charged parallel plates where the uniform field $E = V/d$ exerts a constant force $eE$, deflecting them like a projectile under gravity. Changing the plate voltage steers the beam, the operating principle of older oscilloscopes and televisions.

Example 2. Millikan's oil-drop experiment. A charged oil drop is held stationary between parallel plates when the upward electric force $qE$ balances its weight $mg$. Measuring the balancing voltage gives the charge $q$, and Millikan found charge comes in multiples of the elementary charge $e$, a landmark in physics.

Try this

Q1. Define electric potential at a point. [2 marks]

• Cue. The work done per unit positive charge to bring a small charge from infinity to that point, a scalar with zero at infinity.

Q2. Find the field strength midway between two parallel plates $8.0\ \text{mm}$ apart with $120\ \text{V}$ across them. [2 marks]

• Cue. $E = V/d = 120/(8.0 \times 10^{-3}) = 1.5 \times 10^4\ \text{V m}^{-1}$ (uniform, so the same at every point between the plates).

Q3. State how the electric field strength and the electric potential of a point charge each depend on distance. [2 marks]

• Cue. Field strength $\propto 1/r^2$ (inverse-square); potential $\propto 1/r$ (inverse).