question_answer

The grid (each square of\[1\text{ }m\times 1\text{ }m\]), represents a region in space containing a uniform electric field. If potentials at points O, A, B, C, D, E, F, G, H are respectively \[0,-1,-2,1,2,0,-1,1\] and O volts, find the electric field intensity.

A) \[\left( \hat{i}+\hat{j} \right)V/m\]           

B)        \[\left( \hat{i}-\hat{j} \right)V/m\]

C) \[\left( -\hat{i}+\hat{j} \right)V/m\]

D)        \[\left( -\hat{i}-\hat{j} \right)V/m\]

Correct Answer: B

Solution :

[b] Direction of electric field is perpendicular to equipotential surface and towards decreasing potential. Unit vector of electric field                         \[\overset{\to }{\mathop{E}}\,=\cos 45{}^\circ \hat{i}-\sin 45{}^\circ \hat{j}\] \[\Rightarrow \,\,\,\,\overset{\to }{\mathop{E}}\,=\frac{1}{\sqrt{2}}\,(\hat{i}-\hat{j})\] Magnitude of electric field \[|\overset{\to }{\mathop{E}}\,|\,\,\,=\frac{\Delta V}{\Delta x}=\frac{2}{\sqrt{2}}=\sqrt{2}\,V/m\] Hence, \[\overset{\to }{\mathop{E}}\,=\,|\overset{\to }{\mathop{E}}\,|\,\hat{E}=\frac{1}{\sqrt{2}}(\hat{i}-\hat{j})\times \sqrt{2}=(\hat{i}-\hat{j})\,V/m\]