Result

Field from potential

(E3.4.1) E3.4 · Obtaining the Value of the Electric Field from the Electric Potential →

E_{x} = - \frac{d V}{d x}

Statement

Each field component is minus the slope of V in that direction. E points downhill on the potential landscape.

Symbols and units

$E_{x}$	field component along x	V/m = N/C
$d V / d x$	slope of the potential	V/m

When it applies

V known as a function of position.

Don't use when

Requires V to be known as an explicit, differentiable function of position — a numerically tabulated or discontinuous potential needs a finite-difference or other approximation instead.

Derivation

Take one small step and use the definition of potential difference:

d V = - E \cdot d s

Step along x only, so ds = dx x̂:

d V = - E_{x} d x

Rearrange:

E_{x} = - \frac{d V}{d x}

All three components together make the gradient:

E = - \nabla V

Worked examples

From potential to field

V = 3 x^{2} \Rightarrow E_{x} = - \frac{d V}{d x} = - 6 x

Checked against a point charge:

V = k_{e} \frac{q}{r} \Rightarrow E_{r} = - \frac{d V}{d r} = k_{e} \frac{q}{r ^{2}}

Watch out for

Usually the easy route: compute the scalar V first by adding numbers, then differentiate — much less error-prone than integrating vectors from the start.

Connects to

Potential difference Continuous distribution