Physics - Electrostatics Question with Solution | TestHub

The electric field components are given by $E_x=-\frac{\partial V}{\partial x}$, $E_y=-\frac{\partial V}{\partial y}$, and $E_z=-\frac{\partial V}{\partial z}$. Given $V(x,y,z)=A(xy+yz+zx)$. Calculating the partial derivatives: $E_x=-\frac{\partial }{\partial x}(A(xy+yz+zx))=-A(y+z)$. $E_y=-\frac{\partial }{\partial y}(A(xy+yz+zx))=-A(x+z)$. $E_z=-\frac{\partial }{\partial z}(A(xy+yz+zx))=-A(x+y)$. At the point $(1,1,1)$: $E_x=-A(1+1)=-2A$. $E_y=-A(1+1)=-2A$. $E_z=-A(1+1)=-2A$. The magnitude of the electric field is $E=\sqrt{E_x^2+E_y^2+E_z^2}=\sqrt{(-2A{)}^2+(-2A{)}^2+(-2A{)}^2}=\sqrt{4A^2+4A^2+4A^2}=\sqrt{12A^2}=2A\sqrt{3}$.