An equipotential surface is that at every point of which electric potential is the same.
By definition potential difference between two points B and A = work done in carrying unit positive test charge from A to B. VB – VA = WAB
If points A and B lie on an equipotential surface,
Then VB = VA
∴ wAB = VB – VA = 0
Hence no work is done in moving the test charge form one point of equipotential surface to the other.
If dl is the small distance over the equipotential surface through which unit positive charge is carried then
dW = ~E. ~dl= ~E ~dl cos θ = 0
∴ Cos θ = 0 or θ = 90 0 ~E ⊥ ~dl
∴ Electric field intensity E is always normal to the equipotential surface for any charge configuration equipotential surface through a point is normal to the electric field at that point.
For a single charge q the potential is given by
V = q/4π∈0r
This shows that Vis constant if r is constant hence equipotential surfaces of a single point charge are concentric spherical surfaces centered at the charge as shown in
For a uniform electric field say along the X – axis the equipotential surfaces are planes normal to the X – axis planes parallel to the Y – Z plane
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