How Is Electric Field Strength Related to the Potential Gradient? (HL)

For IB Physics HL, electric field strength is the negative electric potential gradient:

$$\vec{E}=-\nabla V$$

In one dimension, this becomes $E_x=-\frac{\Delta V}{\Delta x}$, or $E_x=-\frac{dV}{dx}$ when the potential changes continuously.

The Reasoning

Electric potential $V$ is the electric potential energy per unit charge. A potential gradient describes how rapidly potential changes with position.

The negative sign means that the electric field points in the direction in which potential decreases most rapidly. A positive test charge released in the field accelerates toward lower potential because the electric force does positive work on it, reducing its electric potential energy.

Since potential is measured in volts and distance in metres, the potential gradient has units $\text{V m}^{-1}$. These are equivalent to the electric field unit $\text{N C}^{-1}$.

More generally, $\vec{E}=-\nabla V$ means the field is perpendicular to equipotential surfaces and directed from higher to lower potential. Moving a charge along an equipotential requires no work because $\Delta V=0$. For a potential-position graph, the electric field component is the negative gradient of the tangent; a steeper graph therefore represents a stronger field. In a uniform field, the graph is a straight line and gives the magnitude between parallel plates. For a non-uniform field, use the local derivative rather than an average gradient.

For example, suppose potential falls from $120\,\text{V}$ to $20\,\text{V}$ over $0.040\,\text{m}$ in the positive $x$-direction:

$$\frac{\Delta V}{\Delta x}=\frac{20-120}{0.040}=-2.5\times10^3\,\text{V m}^{-1}$$

Therefore,

$$E_x=-\frac{\Delta V}{\Delta x}=+2.5\times10^3\,\text{V m}^{-1}.$$

The field points in the positive $x$-direction.

A common misconception is that electric field and electric potential are the same. Potential is a scalar, whereas electric field is a vector determined by the spatial rate of change of potential.

Exam Technique

This is Additional Higher Level (AHL) content in D.2 Electric and magnetic fields. In a calculation, show the potential difference, displacement, negative sign, magnitude, units, and field direction. In an Explain response, state explicitly that the field points toward decreasing potential.