Answer to Question #212829 in Electricity and Magnetism for NICKO

Question

In a certain region of space, the electric potential is V (x, y, z) = Axy − Bx2 + Cy, where A, B, and
C are positive constants, (a) Calculate the x-, y- and z-components of the electric field.(b) At what
point is the electric field equal to zero?

Answer 1

Gives

V(x,y,z)=Axy-Bx²-Cy

$E=-\nabla V$

$E_x=-\frac{dV}{dx}=-(Ay-2Bx)$

$E_y=-\frac{dV}{dy}=-(Ax+c)$

$E_z=-\frac{dV}{dz}=0$

$E=E_x+E_y+E_z$

Find out point where electric field is zero

$$E=-\hat{i}(Ay-2Bx)-\hat{j}(Ax+c)+\hat{k}$$

E=0

$\hat{i} components zero$

Ax+c=0

$x=-\frac{c}{A}$

Ay-2Bx=0

$y=\frac{2Bx}{A}$

Put x value

$y=-\frac{2Bc}{A^2}$

Point $(x,y,z)=(-\frac{c}{A},\frac{2Bc}{A^2},0)$