Determine the magnitude of the electric field in this region

Asked by admin @ 22/11/2021 in Mathematics viewed by 264 People

In a certain region, the electric potential due to a charge distribution is given by the equation v(x,y,z) = 3x2y2 + yz3 - 2z3x, where x, y, and z are measured in meters and v is in volts. calculate the magnitude of the electric field vector at the position (x,y,z) = (1.0, 1.0, 1.0).

Answered by admin @ 22/11/2021

The electric potential due to a charge distribution is given by

$V(x,y,z)=3x^{2}y^{2}+yz^{3}+2z^{3}x$ (Equation 1)

Differentiating the above equation 1 with respect to x,

$V_{x}=6xy^{2}+2z^{3}$ (Equation 2)

Differentiating the above equation 1 with respect to y,

$V_{y}=6x^{2}y+z^{3}$ (Equation 3)

Differentiating the above equation 1 with respect to z,

$V_{z}=3yz^{2}+6z^{2}x$ (Equation 4)

Substituting the value of x,y and z in equations 2,3 and 4

$V_{x}=6+2=8$

$V_{y}=6+1=7$

$V_{z}=3+6=9$

Magnitude = $\sqrt{8^{2}+7^{2}+9^{2}}$

= $\sqrt{194}$

= 14.0

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