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Formulas/physics/Electrostatic Potential/Work Done Moving a Charge

Work Done Moving a Charge

Work done by the electric field in moving charge q from A to B. Depends only on endpoints — not on the path. This is the statement that the electrostatic force is conservative.
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Derivation

Derivation

Work done by the electric field in moving charge qq from A to B:

WA→B=q∫ABE⃗⋅dr⃗=−q∫ABdV=−q[V]AB=−q(VB−VA)W_{A\to B} = q\int_A^B \vec{E}\cdot d\vec{r} = -q\int_A^B dV = -q[V]_A^B = -q(V_B - V_A) W=q(VA−VB)\boxed{W = q(V_A - V_B)}

Path independence

Since E⃗=−∇V\vec{E} = -\nabla V, the electrostatic field is conservative. The work depends only on VAV_A and VBV_B — not on the path taken from A to B.

Sign convention

  • VA>VBV_A > V_B (moving with the field): W>0W > 0, field does positive work, KE increases
  • VA<VBV_A < V_B (moving against the field): W<0W < 0, field does negative work, PE increases
  • For a positive charge: motion from high V to low V is spontaneous (like a ball rolling downhill)
  • For a negative charge: spontaneous motion is from low V to high V

Work by external agent

Work done by the external agent (against the field):

Wext=q(VB−VA)=−WfieldW_{ext} = q(V_B - V_A) = -W_{field}

This equals the change in potential energy: Wext=ΔU=qΔVW_{ext} = \Delta U = q\Delta V.

Remember
The electron volt (eV) is defined from this formula: $1\,\text{eV}$ is the work done moving one electron through a potential difference of 1 V. $1\,\text{eV} = 1.6\times10^{-19}\,\text{J}$.