Find the potential difference `V_(AB)` between `A (0,0,0) and B (1 m , 1 m,1 m)` in an electric field :
(i) `vec E = (y hat i + x hat j) Vm^-1`
(ii) `vec E = (3 x^2 y hat i + x^3 hat j) Vm^-1`.

Explanation

(i) As `dV = - vec E. vec d r`
Here `vec E = (y hat i + x hat j)Vm^-1 and vec d r = d x hat i + d y hat j
+ d z hat k`
Hence, `dV = -(y hat i + x hat j).(d x hat i + d y hat j + d z hat k)`
`= - y d x + x d y = - d (x y)`
[Using multiplication rule of derivative]
`V_(AB) = - underset(((1, 1, 1)))overset(((0, 0, 0)))(int) d(x y) = - [xy]_(((1, 1, 1)))^(((0, 0, 0))) = 1 V`
(ii) As `dV = int vec E . vec d r`,
`vec E = -(3 x^3 y hat i + x^3 hat j)` and `vec d r = d x hat i + d y hat j + d z hat k`
`vec d V = -(3 x^2 y hati + x^3 hat j). (d x hat i + d y hat j + d z hat k)`
Hence, `dV = -(3 x^2 y d x + x^3 d y) = - d (x^3 y)`
[Using multiplication rule of differentiation]
`V_(AB) = - underset(((1, 1, 1)))overset(((0, 0, 0)))(int) d(x^3 y) = - [x^3 y] _(((1,1,1)))^(((0,0,0))) = 1 V`.