Question

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

• A.
• B.
• C.
• D.

Answer 1

Answer: (B)

Answer 2

Electric Field Due to an Infinitely Long Charged Wire:

For an infinitely long line of charge with linear charge density $\lambda$, the electric field $E$ at a distance $r$ from the wire is given by:

$E = \frac{2k\lambda}{r}$
where $k$ is Coulomb’s constant.

Centripetal Force on the Electron:

The electron revolves in a circular path due to the centripetal force provided by the electric field. The centripetal force $F$ acting on the electron of charge $e$ is:

$F = eE = e \times \frac{2k\lambda}{r} = \frac{2ke\lambda}{r}$
This force provides the necessary centripetal force for the electron’s circular motion, which is given by:

$F = \frac{mv^2}{r}$ where $m$ is the mass of the electron and $v$ is its velocity.

Kinetic Energy of the Electron:

Equating the expressions for the centripetal force:

$\frac{mv^2}{r} = \frac{2ke\lambda}{r}$ Simplifying, we get:

$mv^2 = 2ke\lambda$

The kinetic energy $KE$ of the electron is:

$KE = \frac{1}{2}mv^2 = \frac{1}{2} \times 2ke\lambda = ke\lambda$

Notice that the kinetic energy $KE$ is independent of $r$ and remains constant as $r$ changes.

Conclusion:

Since the kinetic energy of the electron does not depend on the radius $r$, the correct graph showing the kinetic energy as a constant with respect to $r$ is Option (2).