Question

Over a certain region of space, the electric potential is V = 2x - 5x²y + 2yz². Find the expression for the x component of the electric field over this region. (Use the following as necessary: x, y, and z.)

E_x =  

Find the expression for the y component of the electric field over this region.

E_y =   

Find the expression for the z component of the electric field over this region.

E_z =  

What is the magnitude of the field at the point P, which has coordinates (2, 0, -4) m?
N/C

Answer 1

The relation between electric field (E) and electric potential (V) is given by

$\dpi{100} \vec{E} = - \, grad \, (V) = - \left ( \hat{i} \, \frac{\partial }{\partial x} + \hat{j} \, \frac{\partial }{\partial y} + \hat{k} \, \frac{\partial }{\partial z} \right ) V$

Given,

  $\dpi{100} V = 2x-5x^{2}y + 2yz^{2}$

So,

$\dpi{100} \vec{E} = - \, grad \, (V) = - \left ( \hat{i} \, \frac{\partial }{\partial x} + \hat{j} \, \frac{\partial }{\partial y} + \hat{k} \, \frac{\partial }{\partial z} \right ) (2x-5x^{2}y + 2yz^{2})$

     $\dpi{100} = - \left ( \hat{i} \, \frac{\partial (2x-5x^{2}y + 2yz^{2})}{\partial x} + \hat{j} \, \frac{\partial (2x-5x^{2}y + 2yz^{2})}{\partial y} + \hat{k} \, \frac{\partial (2x-5x^{2}y + 2yz^{2})}{\partial z} \right )$

  $\dpi{100} = - \left ( \hat{i} \, (2 - 10 xy)+ \hat{j} \, (-5x^{2} + 2z^{2})+ \hat{k} \, (4yz) \right )$

  $\dpi{100} = \hat{i} \, ( 10 xy - 2)+ \hat{j} \, (5x^{2} - 2z^{2})+ \hat{k} \, (-4yz)$

We know that  

$\dpi{100} \vec{E} = \hat{i} \, E_{x} + \hat{j} \, E_{y} + \hat{k} \, E_{z}$

Comparing these two equations we get ,

$\dpi{100} E_{x} = 10 xy -2$

  $\dpi{100} E_{y} = 5x^{2} - 2z^{2}$

$\dpi{100} E_{z} = -4yz$

At point P(2 , 0 , -4) m,

$\dpi{100} E_{x} = 10 \times 2 \times 0-2 = -2$ N/C

$\dpi{100} E_{y} = 5(2)^{2} - 2(-4)^{2} = 20 - 32 = -12$ N/C

$\dpi{100} E_{z} = -4\times 0 \times (-4) =0$ N/C

So, magnitude of field at point P is

$\dpi{100} E = \left | \vec{E} \right | = \sqrt{E_{x}^{2} + E_{y}^{2} + E_{z}^{2}}$

  $\dpi{100} = \sqrt{(-2)^{2} + (-12)^{2} + (0)^{2}} = \sqrt{148} = 12.165 \, \, N/C$

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