The electirc potential at a point `(x, y, z)` is given by
`V = -x^(2)y - xz^(3) + 4`
The electric field `vecE` at that point is

Options

A. `vec(E )= hat(i) 2xy + hat(j) (x^(2) + y^(2)) + hat(k) (2xy- y^(2))`
B. `vec(E )= hat(i) z^(3) + hat(j) xyz + hat(k) z^(2)`
C. `vec(E )= hat(i) (2xy+ z^(3)) + hat(j) xy^(2) + hat(k) 3z^(2)x`
D. `vec(E )= hat(i) (2xy + z^(3)) + hat(j) x^(2) + hat(k) 3xz^(2)`

Answer

D

Explanation

`v= -x^(2)y - xz^(2) + 4`
`vec(E )= (-del v)/(dx) hat(i)- (-del v )/(dy) hat(j)- (-del v)/(dz) hat(k)`
`vec(E )_(s)= (-del v)/(dx) hat(i)= - [(del)/(del x) (-x^(2)y- xz^(3) +4)] hat(i)`
`=(2xy + z^(3)) hat(i)`
`vec(E )= (-del v)/(dx) hat(j)= (-del)/(dy) (x^(2)y- xz^(3) + 4) hat(j)`
`=x^(2)hat(j)`
`vec(E )_(s)= +3xz^(2)hat(k)`
So, `vec(E )= (2xy + z^(3)) hat(i) + x^(2) hat(j) +3xz^(1) hat(k)`