Question

In a certain region of space, the electric potential is $V(x,y,z)=Axy-B{x}^2$ $+Cy$, where $A,BandC$ are positive constants. At which points is the electric field equal to zero?

• A. $x=+C/A,y=+BC/A$ ${}^2$, any value of $z$
• B. $x=+C/A,y=+2BC/A$ ${}^2$, any value of $z$
• C. $x=-C/A,y=-2BC/A$ ${}^2$, $z=0$
• D. $x=-C/A,y=-2BC/A$ ${}^2$, any value of $z$

Answer 1

The correct option is D $x=-C/A,y=-2BC/A$ ${}^2$, any value of $z$

$\overrightarrow{E}=-▽V=-\left[\right(Ay-2Bx)\hat{i}+(Ax+C\left)\hat{j}\right]$

$\overrightarrow{E}=0$ at,

$E_y=0\Rightarrow Ax+C=0\Rightarrow x=-C/A$

$E_x=0\Rightarrow Ay-2Bx=Ay+2BC/A=0\Rightarrow y=-2BC/A^2$

$E_z=0$ everywhere .