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

The Correct Option is B

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).