# The electric potential is V= 5x − 3x^2y+2yz. Find the expressions for the x, y, and z components of the electric field over this region. What is the magnitude of the field at the point P that has coordinates (1, 0, -2) m?

Jul 27, 2018
• $\boldsymbol{E} = \left(\begin{matrix}- 5 x + 6 x y \\ 3 {x}^{2} - 2 z \\ - 2 y\end{matrix}\right)$

• ${\left\mid \boldsymbol{E} \right\mid}_{\text{(1,0,-2)}} = \sqrt{60}$

#### Explanation:

• Part 1

$\boldsymbol{E} = - \nabla V$

= - ((del_x(5x − 3x^2y+2yz) ),(del_y(5x − 3x^2y+2yz) ),(del_z(5x − 3x^2y+2yz)) )

$= \left(\begin{matrix}- 5 + 6 x y \\ 3 {x}^{2} - 2 z \\ - 2 y\end{matrix}\right)$

• Part2

$\boldsymbol{E} \left(1 , 0 , - 2\right) = \left(\begin{matrix}- 5 \\ 7 \\ 0\end{matrix}\right)$

$\left\mid \boldsymbol{E} \right\mid = \sqrt{{\left(- 5\right)}^{2} + {7}^{2} + {0}^{2}} = \sqrt{74}$