Question

The electric potential existing in space is $\hspace{0.17em}V(x,y,z)=A(xy+yz+zx).$ (a) Write the dimensional formula of A. (b) Find the expression for the electric field. (c) If A is 10 SI units, find the magnitude of the electric field at (1 m, 1 m, 1 m).

Answer 1

Given:
Electric potential, $V(x,y,z)=A(xy+yz+zx)$
$$\begin{gathered} A=\frac{\mathrm{volt}}{{\mathrm{m}}^2} \\ \Rightarrow \left[A\right]=\frac{\left[{\mathrm{ML}}^2{\mathrm{I}}^{-1}{\mathrm{T}}^{-3}\right]}{\left[{\mathrm{L}}^2\right]} \\ \Rightarrow A=\left[{\mathrm{MT}}^{-3}{\mathrm{I}}^{-1}\right] \end{gathered}$$

(b) Let E be the electric field.
$$\begin{gathered} dV=-\overrightarrow{E}.\overrightarrow{dr} \\ \Rightarrow A(y+z)dx+A(z+x)dy+A(x+y)dz=-E(dx\hat{i}+dy\hat{j}+dz\hat{k}) \\ \Rightarrow \left[A\right(y+z)\hat{i}+A(z+x)\hat{j}+A(x+y\left)\hat{k}\right][dx\hat{i}+dy\hat{j}+dz\hat{k}]=-E\left[dx\hat{i}+dy\hat{j}+dz\hat{k}\right] \end{gathered}$$
Equating now, we get
$\overrightarrow{E}=-A(y+z)\hat{i}-A(z+x)\hat{j}-A(x+y)\hat{k}$

(c) Given: A = 10 V/m²
$$\begin{gathered} r=(1\mathrm{m},1\mathrm{m},1\mathrm{m}) \\ \overrightarrow{E}=-10\left(2\right)\hat{i}-10\left(2\right)\hat{j}-10\left(2\right)\hat{k} \\ =-20\hat{i}-20\hat{j}-20\hat{\mathrm{k}} \end{gathered}$$
Magnitude of electric field,
$$\begin{gathered} \left|E\right|=\sqrt{{20}^2+{20}^2+{20}^2} \\ =\sqrt{1200}=34.64=35\mathrm{N}/\mathrm{C} \end{gathered}$$