The electric potential existing in space is $V (x, y, z) = A (x y + y z + z x)$ . Find the expression for the electric field.

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Solution

The electric potential in the space is given as:
$V (x, y, z) = A (x y + y z + z x)$

The electric field in the region is given as:
$E = - \frac{δ V^i}{δ x} - \frac{δ V^j}{δ y} - \frac{δ V^k}{δ z}$

$= - [\frac{δ}{δ x} [A (x y + y z + z x) + \frac{δ}{δ y} [A (x y + y z + z x) + \frac{δ}{δ z} [A (x y + y z + z x)]$

$= - [(A y + A z)^i + (A x + A z)^j + (A y + A z)^k]$

$= - A (y + z)^i + A (x + z)^j + A (y + x)^k$ .

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