Edition Solution - Elements Of Electromagnetics Sadiku 7th

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Problem Statement: Given the electric field $\mathbfE = (x^2 + y^2)\mathbfa_x + (y^2 - z^2)\mathbfa_y + (z^2 - x^2)\mathbfa_z$, find the electric potential $V$ at point $(1, 1, 1)$ with respect to the origin, assuming $V = 0$ at the origin. By the end of the semester, you will

$\frac\partial V\partial z = 2zy + \fracdg(z)dz = z^2 - x^2$, implying $\fracdg(z)dz = z^2 - x^2 - 2zy$.

In Cartesian coordinates, $\nabla V = \frac\partial V\partial x\mathbfa_x + \frac\partial V\partial y\mathbfa_y + \frac\partial V\partial z\mathbfa_z$. Therefore, $\mathbfE = -\frac\partial V\partial x\mathbfa_x - \frac\partial V\partial y\mathbfa_y - \frac\partial V\partial z\mathbfa_z$.

To find $V$ at $(1,1,1)$ with respect to the origin, use the line integral $V = -\int_0^1,1,1 \mathbfE \cdot d\mathbfl$.