If potential (in volts) in a region is expressed as `V (x, y, z) = 6xy - y + 2yz`, the electric field (in `N//C)` at point `(1, 1, 0)` is

Options

A. `-(6 hat(i) + 9 hat(j) + hat(k))`
B. `-(3 hat(i) + 5 hat(j) + 3 hat(k))`
C. `-(6 hat(i) + 5 hat(j) + 2 hat(k))`
D. `-(2 hat(i) + 3 hat(j) + hat(k))`

Answer

C

Explanation

Given, `V = 6 xy-y +2 yz`
`vec(E) = [(del V)/(del x) hat(i) + (del V)/(del y) hat(j) + (del V)/(del z) hat(k)]`
`vec(E) = -{(del)/(del x) [6 xy - y + 2 yz] hat(i) + (del)/(del y) [6xy - y +2yz] hat(j)`
`+ (del)/(dz) [ 6 xy - y+ 2 yz) hat(k) }`
`= -{(6y) hat(i) + (6x - 1 + 2z) hat(j) +(2y) hat(k)}`
`vec(E)_(1,1,0) = -{(6x1) hat(i) + (6xx1-1+2xx0) hat(j) + (2xx1) hat(k)]`
`= -(6 hat(i) + 5 hat(j) + 2 hat(k))`