Work Done in Moving a Charge in an Electric Field

An electric field can do work on a charge as it moves from one point to another. This work depends only on the charge and the potential difference between the start and end points — not on the shape of the path taken. This is because the electrostatic field is a conservative field, much like gravity.

Work done in moving a charge in an electrostatic field is the energy transferred when a charge q is displaced between two points having a potential difference V_A − V_B.

W = q(V_A​ − V_B​)

Where:

• W = work done (joules)
• q = magnitude of charge (coulombs)
• V_A, V_B​ = electric potentials at points A and B (volts), with A at higher potential

1. Consider a uniform electric field E directed from A to B.
2. Potential difference between A and B: V_A − V_B = Ed, where d is the separation along the field direction.
3. Substitute into the general formula: W = q(VA − VB) = q(Ed)
4. Final special-case formula: W = qEd

Remember: W = qEd is valid only for a uniform field, while W = q(V_A − V_B) is the general formula valid everywhere.

Analogy: Think of potential like height on a hill. A positive charge moving from high to low potential is like a ball rolling downhill — the field (gravity) does the work naturally. Moving it uphill needs an external push (external work).

• Electrostatic force is a conservative force.
• Work done depends only on the initial and final positions, not on the path taken.

• Capacitors - charge storage relies on work done against/by the field between plates
• Batteries - internal work moves charges against the potential gradient during charging
• Cathode-ray tubes/particle accelerators - electrons gain kinetic energy from work done by the field