Determine the electric field strength vector if the potential of this field depends upon x - and y - coordinates as :
(i) `V = a(x^2 - y^2)`
(ii) `V = a x y`.

Explanation

If `u = fn(x)`
`(del u)/(del x) = (du)/(dx)`
(a) `V = - 4x^(2)`
`E_(x) = -(dV)/(dx) = -(8x) = 8x`
`vec(E ) = E_(x) hat(i) = 8 x hat(i)`
(b) `V = k(x^(2) - y^(2))`
`E_(x) = -(del V)/(del x) = -k . 2x = -2 kx`
`E_(y) = -(del V)/(del y) = - k.(-2y) = 2 ky`
`vec(E) = E_(x) hat(i) + E_(y) hat(j) = -2k(x hat(i) - y hat(j))`
(c ) `V = k x y`
`E_(x) = -(del V)/(del x) = -k y`
`E_(y) = -(del V)/(del y) = - k x`
`vec(E ) = E_(x) hat(i) + E(y) hat(j) = -k(y hat(i) + x hat(j))`
(d) `V = - k(xy + yz + zx)`
`E_(x) = -(del V)/(del x)= k (y + z)`
`E_(y) = -(del V)/(del y) = k(x + z)`
`E_(z) = -(del V)/(del z) = k( y + x)`
`vec(E) = E_(x) hat(i) + E_(y) hat(j) + E_(z) hat(k) = k[(y + x)hat(i) + (x + z)hat(j) + (y + x) hat(k)]`