D.2.15 (HL)—Electric potential gradient

Use the gradient relationship

Electric field strength is the negative spatial gradient of electric potential. On a potential–distance graph, use the tangent gradient at the required point and state whether the answer is a signed component or a magnitude.

Worked example — tangent gradient

If a tangent changes from $-26\,\mathrm{kV}$ to $0\,\mathrm{V}$ over $8.0\,\mathrm{cm}=8.0\times10^{-2}\,\mathrm{m}$, $E=-[0-(-26\times10^3)]/(8.0\times10^{-2})=-3.3\times10^5\,\mathrm{V\,m^{-1}}$. The negative sign gives the field direction in the chosen coordinate.

Interpret the sign

The negative sign means the electric field points toward decreasing potential. A negative slope of $V_e$ against position corresponds to a positive field component in that coordinate direction; state whether the question wants a signed component or a magnitude.

Connect field to motion

A negative charge experiences force opposite to the electric field. Therefore its acceleration direction is opposite to the direction of decreasing potential, even though the field itself is always defined using a positive test charge.

Common trap

Do not use the graph’s potential value instead of its local gradient, and do not reverse the particle’s force direction without checking the particle’s charge sign.