In a certain region,the electric potential due to a charge distribution is given by the equation V(x,y)= 2xy - x² - y,where x and y are measured in meters and V is in volts.At which point is the electric field equal to zero? A) x = 0.5 m, y = 1 m B) x = 1 m, y = 1 m C) x = 1 m, y = 0.5 m D) x = 0.5 m, y = 0.5 m E) x = 0 m, y = 0 m

Question

Quizplus · Accepted Answer

The electric field is given by the negative gradient of the electric potential:$$\overrightarrow{E}=-
abla V = -\left(\frac{\partial V}{\partial x}\hat{x} + \frac{\partial V}{\partial y}\hat{y}ight)$$So, the electric field is zero when both partial derivatives are zero:$$\frac{\partial V}{\partial x} = 2y - 2x = 0$$$$\frac{\partial V}{\partial y} = 2x - 1 = 0$$Solving these equations simultaneously gives:$$x = 0.5 	ext{ m}$$$$y = 0.5 	ext{ m}$$Therefore, the electric field is zero at the point (0.5 m, 0.5 m).

In a Certain Region,the Electric Potential Due to a Charge