Question

In a uniform electric field, the potential is $10V$ at the origin of coordinates, and $8V$ at each of the points $(1,0,0),(0,1,0)$ and $(0,0,1)$. The potential at the point $(1,1,1)$ will be

• A. $0$
• B. $4V$
• C. $8V$
• D. $10V$

Answer 1

The correct option is B $4V$

Relation between potential and electric field
$dV=-\overrightarrow{E}.\overrightarrow{dr}.....\left(1\right)$

Let
$\overrightarrow{E}=E_x\hat{i}+E_y\hat{j}+E_z\hat{k}$
$\overrightarrow{dr}=dx\hat{i}+dy\hat{j}+dz\hat{k}$

Apply equation (1) between $(0,0,0)$ and $(1,0,0)$
${\int }_{Vat(0,0,0)}^{Vat\left(1,0,0\right)}dV=-E{\int }_{0,0,0}^{1,0,0}dr{V}_{(1,0,0)}-V_{(0,0,0)}=-E_x\times 1$
$8-10=-E_x$
$E_x=2$
Apply equation (1) between $(0,0,0)$ and $(0,1,0)$
${\int }_{Vat(0,0,0)}^{Vat\left(0,1,0\right)}dV=-E_y{\int }_{0,0,0}^{1,0,0}dr{V}_{(0,1,0)}-V_{(0,0,0)}=-E_y\times 1$
$8-10=-E_y$
$E_y=2$
Apply equation (1) between (0,0,0) and (0,0,1)
${\int }_{Vat(0,0,0)}^{Vat\left(0,0,1\right)}dV=-E_z{\int }_{0,0,0}^{1,0,0}dr{V}_{(0,0,1)}-V_{(0,0,0)}=-E_z\times 1$
$8-10=-E_z$
$E_z=2$


So,
$\overrightarrow{E}=2(\hat{i}+\hat{j}+\hat{k})$
Apply equation (1) between $(0,0,0)$ and $(1,1,1)$
${\int }_{Vat(0,0,0)}^{Vat\left(1,1,1\right)}dV=-{\int }_{0,0,0}^{1,1,1}\overrightarrow{E}.\overrightarrow{dr}{V}_{(1,1,1)}-V_{(0,0,0)}=-[2x{|}_0^1+2y{|}_0^1+2z{|}_0^1$
$V_{(1,1,1)}-10=-6$
$V_{(1,1,1)}=10-6=4$