ln a certain region, the electric potential is given by the formula V (x, y, z) = `2x^(2)y + 3y^(3)z - 4z^(4)x`. Find the components of electric field and the vector electric field at point (1, 1, 1) in this field.

Explanation

V(x,y,z) = `2x^(2) y + 3y^(3)y+3y^(3)z -4z^(4)x`
`:. E_(x)=- (delV)/(delx)`
`=- (del)/(delx)(2x^(2)y + 3y^(3)z - 4z^(4)x)`
`:. E_(x) = -(4xy+0-4z^(4))`
Putting x =1 , y =1 , z=1
`E_(x) = -(4-4) `
`E_(x) = 0` and `E_(y) =-(delV)/(dely)`
`=- (del)/(dely) ( 2x^(2)y+3y^(3) z-4z^(4)x)`
`:. E_(y) =- (2x^(2)+9y^(2)z-0)`
`:. E_(y) = -(2x^(2)+9y^(2)z)`
Putting x =1 , y =1 , z=1
`:. E_(y) = -(2+9) =-11 ` and `E_(z) =-(del)/(delz) (2x^(2)y+2y^(3) z-4z^(4)x)`
`= - (0+3y^(3)-16z^(3)x)`
Putting x =1 , y=1 , z=1
`E_(z) = (0+3-16) =13`
Putting the value of components in equation
`vecE=Exhati+Eyhatj+Ezhatk`
`=0hati-11hatj-13hatk`
`= - 11hatj +13hatk`