Prove that $\vec{E}$ is always strictly perpendicular to Equipotential surfaces.
Key Equations
\( \vec{E} = -\nabla V \)
\( \Delta V = -\int \vec{E} \cdot d\vec{s} \)
Point Charge:
\( V = \frac{kq}{r} \)
Why It Matters
Equipotential mapping transforms a complex vector field into an intuitive topographical map of electrical "elevation." Moving along a line costs $0$ Joules of work. Moving across lines requires integrating the electric field.
Interactive Gaussian Graph Paper
Under Cursor ($V$)0.0 V
Field Mag ($|\vec{E}|$)0.0 V/m
Click anywhere to draw an infinite Equipotential Line at that voltage contour.
Charge Configuration
Display Options
Quick Quiz
Why is the Electric Field exactly zero exactly midway between two IDENTICAL positive charges?