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:
The Correct Option is B
Solution and Explanation
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).
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