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

(a) `dV = -(E_(x) dx + E_(y) dy) = -(ydx + xdy)`
`int y dx = yx` or `int xdy = xy`
Since integer of both terms is same i.e. `vec(E ) . D vec(r )` is a perfect differential.
`V_(1,2,3) - V_(0,0,0) = - int_(0,0,0)^(1,2,3) vec(E ) . d vec(r ) = - int(E_(x) dx + E_(y) dy)`
`= - int (y dx + xdy) = - int d(xy) = -xy`
`= -|xy|_(0 ,0)^(1,2)`
`= - [ (1 xx 2) - (0 , 0)] = -2`
`= -2` volt
(b) `dV = -(E_(x) dx + E_(y) dx)`
` = -(3 x^(2) y dx + x^(3) dy)`
`int3x^(2) y dx = y int 3 x^(2) dx = y . 3 (x^(3))/(3) = x^(3) y`
`int x^(3) dy = x^(3) int dy = x^(3) y`
Since integral of both terms is same , i.e. `vec(E ) . d vec(r)` is perfect differential.
`V_(1,2,3) - V_(0,0,0) = - int_(0,0,0)^(1,2,3) (3x^(2) y dx + x^(3) dy) = - int_(0,0,0)^(1,2,3) d(x^(3) y)`
`= - [x^(3) y ]_(0,0)^(1,2) = - [(1^(3) xx 2) - 0 xx 0)]`
` = - 2` volt