Electric Field as Gradient of Electric Potential: Relation between E and V

Electric field and electric potential are two ways of describing the same electrostatic situation — one in terms of force per unit charge, the other in terms of energy per unit charge. This topic connects the two using a single elegant mathematical relation.

Analogy: Think of electric potential like the height of a hill and electric field like its slope. The steeper the slope (faster the height drops), the stronger the field. Just as water flows from high to low ground, the electric field points from high to low potential.

Step 1: Consider a small positive test charge q₀​ moved a small distance dr against the electric field E.

Step 2: Work done by an external agent against the field: dW = −q0E dr

Step 3: By definition, potential difference is work done per unit charge: dV = \[\frac {dW}{q_0}\]

Step 4: Combining Steps 2 and 3: dV = −E dr

Step 5: Rearranging gives the final relation: E = −\[\frac {dV}{dr}\]

1 NC⁻¹ = 1 Vm⁻¹

This confirms that electric field can be expressed either in newtons per coulomb (force-based) or volts per metre (potential-based).

Case A: Uniform Field Between Parallel Plates

For two plates separated by a distance d with a potential difference V₁ − V_2​:

A parallel plate capacitor has a potential difference of 50 V across plates 2 mm apart. Find the electric field.

Solution: