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

Answer

D

Explanation

The electric field `vec(E)` and potential V in a region are related as
`vec(E) = -[(del V)/(del x) hat(i) + (del V)/(del y)hat(i) + (del V)/(del z)hat(k)]`
Here, V(x,y,z) = 6xy - y + 2yz
`:. vec(E) = -[(del)/(del x)(6xy - y + 2yz)hat(i) + (del)/(del y)(6xy - y + 2yz) hat(j) + (del)/(del z)(6xy - y + 2yz)hat(k)]`
`= - [(6y)hat(i) + (6x-1 + 2z) hat(k) + (2y)hat(k)]`
At point (1,1,0),
`vec(E) = -[(6(1))hat(i) + (6(1)-1 + 2(0)) hat(j) + (2(1))hat(k)]`
`= -(6hat(i) + 5hat(j) + 2hat(k))`