The equation of an equipotential line in an electric field is $y=2x$, then the electric field strength vector at $(1,2)$ may be :

The correct option is D $-8\hat{i}+4\hat{j}$
The equation of equipotential line,
$y=2x$

So, slope of equipotential line, $m_1={\left(\frac{dy}{dx}\right)}_1=2$

We know that, Electric field along the equipotential surface should be zero.

Since, net electric field should be perpendicular to a equipotential surface.

Equation that represents electric field line will have slope $m_2=-\frac{1}{2}$ $(\because m_1{m}_2=-1)$

But slope , $m_2=\tan \theta =\frac{y}{x}$

From the given options, $\frac{y}{x}$ value becomes $-\frac{1}{2}$ only for option (d).

Hence, option (d) is the correct answer.

Why this question? It intends to test your understanding of relation between slopes of two mutually perpendicular straight lines. since equipotential line is perpendicular to electric field line we can say that, $m_1{m}_2=-1$