An infinitely long positively charged straight thread has a linear charge density $$\lambda\ Cm^{-1}$$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is:

Electric field due to an infinite line charge:

$$E=\frac{\lambda}{2\pi\epsilon_0r}$$

Force on electron provides centripetal force:

$$eE=\frac{mv^2}{r}$$

Substitute E:

$$e\cdot\frac{\lambda}{2\pi\epsilon_0r}=\frac{mv^2}{r}$$

r cancels:

$$\frac{e\lambda}{2\pi\epsilon_0}=mv^2$$

So

$$v^2=\frac{e\lambda}{2\pi\epsilon_0m}$$

which is constant.

Kinetic energy:

$$K=\frac{1}{2}mv^2$$

Also constant, independent of r.

so its constant