If potential (in volts) in a region is expres | vedclass

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Question

(C) Given the potential function: $V = 6xy - y + 2yz$.The electric field $\vec{E}$ is related to the potential $V$ by the relation: $\vec{E} = -
abla

Vedclass · Accepted Answer

$-(6\hat{i} + 5\hat{j} + 2\hat{k})$

Vedclass · Answer

(C) Given the potential function: $V = 6xy - y + 2yz$.The electric field $\vec{E}$ is related to the potential $V$ by the relation: $\vec{E} = -
abla V = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} ight)$.Calculating the partial derivatives:$\frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(6xy - y + 2yz) = 6y$$\frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(6xy - y + 2yz) = 6x - 1 + 2z$$\frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(6xy - y + 2yz) = 2y$Substituting these into the expression for $\vec{E}$:$\vec{E} = -[ (6y)\hat{i} + (6x - 1 + 2z)\hat{j} + (2y)\hat{k} ]$.Now,evaluating at the point $(1, 1, 0)$:$\vec{E}_{(1, 1, 0)} = -[ (6 	imes 1)\hat{i} + (6 	imes 1 - 1 + 2 	imes 0)\hat{j} + (2 	imes 1)\hat{k} ]$$\vec{E}_{(1, 1, 0)} = -(6\hat{i} + 5\hat{j} + 2\hat{k}) \ N/C$.

If potential (in volts) in a region is expressed as $V(x, y, z) = 6xy - y + 2yz$,the electric field (in $N/C$) at point $(1, 1, 0)$ is:

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