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Given the potential $V=2x^2 y-5zx$ and a point P(-4 , 3 , 6). Find V, E, D and $\rho_v$ at P.
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To find V at P:

$V |_{(-4 ,3,6)}=2 (-4)^2 (3)- 5(6)(-4)= 216 v$

To find $\bar{E}$:

$\bar{E}= -∇ v= -\bigg[\dfrac{dv}{dx} \overline{a_x}+\dfrac{dv}{dy} \overline{a_y}+\dfrac{dv}{dz} \overline{a_z} \bigg] \\ \bar{E}= -[(4 ×y-5z) \overline{a_x}+ (2×2) \overline{a_y}+ (-5x) \overline{a_z}] \\ \bar{E}= -4xy+5z \overline{a_x }- 2x^2 \overline{a_y}+ 5x \overline{a_z} \\ \bar{E} |_{(-4,3,6 )}= -4(-4)(3)+ 5(6) \overline{a_x }- 2 (-4)^2 a_y+ 5(-4) \overline{a_z } \\ \hspace{1.5cm} = 78 \overline{a_x }- 32 a_y- 20 \overline{a_z } $

To find $\bar{D}$:

$\bar{D}= ε_o \bar{E} \\ \bar{D}= ε_o [78 \overline{a_x }- 32 a_y- 20 \overline{a_z } ] \\ \bar{D} |_{(-4,3,6)}= ε_o ×E |_{(-4,3,6)}$

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