# In a region, the potential is represented by V(x,y,z) = 6x - 8xy - 8y + 6yz, where V is in volts and x, y, x are in metres. The electric force experienced by a charge of 2 coulomb situated at point (1, 1, 1) is: Option 1) $6\sqrt{5}$ N Option 2) 30 N Option 3) 24 N Option 4) 4$\sqrt 35$ N

D Divya Saini

In space -

$\dpi{100} E_{x}=\frac{-dv}{dx}$  ,  $\dpi{100} E_{y}=\frac{-dv}{dy}$    ,  $\dpi{100} E_{z}=\frac{-dv}{dz}$

-

$V \left ( x,y,z \right )=6x-8xy-8y+6yz$

$\vec{E} = - \vec{V}V = \frac{\vartheta\vec{V}}{\vartheta x}\hat{j}-\frac{\vartheta\vec{V}}{\vartheta y}\hat{j}-\frac{\vartheta\vec{V}}{\vartheta z}\hat{k}$

$\vec{E} = \left ( 6-8y \right )\hat{j} -\left ( -8x-8-6z \right )\hat{j}-\left ( 6y \right )\hat{k}$

$= -\left ( 6-8y \right )\hat{j}+\left ( 8x+8-6z \right )\hat{j} -6y\hat{k}$

$\vec{E} = 2\hat{j}+10\hat{j}-6 \;\hat{k}$

$\begin{vmatrix} \bar{E} \end{vmatrix}=\sqrt{2^{2}}+10^{2}+6^{2}=\sqrt{140}=2\sqrt{35}$

$q\begin{vmatrix} \bar{E} \end{vmatrix}= 4\sqrt{35} N$

Option 1)

$6\sqrt{5}$ N

Incorrect

Option 2)

30 N

Incorrect

Option 3)

24 N

Incorrect

Option 4)

4$\sqrt 35$ N

Correct

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