Question

An electron is released in the field generated by a non-conductivity sheet of uniform surface charge density $ \sigma $. The rate of change of de-Broglie wavelength associated with the electron varies inversely as the $ n $th power of the distance travelled. Find the value of $ n $.

• A. \( \frac{1}{2} \)
• B. 2
• C. \( \frac{1}{4} \)
• D. 4

Hint:

In problems involving electric fields and de-Broglie wavelengths, the rate of change of wavelength is often inversely related to the square root of the distance, which follows from the momentum relation and the forces acting on the particle.

Answer 1

The Correct Option is A

We are given that the rate of change of the de-Broglie wavelength \( \lambda \) of the electron varies inversely with the \( n \)th power of the distance travelled.
We need to find the value of \( n \). The de-Broglie wavelength \( \lambda \) of an electron is given by: \[ \lambda = \frac{h}{p} \] where \( h \) is Planck's constant and \( p \) is the momentum of the electron. The momentum of the electron in the electric field of the sheet is influenced by the force exerted on the electron by the sheet, which depends on the electric field created by the sheet.
The electric field \( E \) generated by the uniformly charged sheet is: \[ E = \frac{\sigma}{2\epsilon_0} \] The force \( F \) acting on the electron is: \[ F = eE = e \frac{\sigma}{2\epsilon_0} \] The rate of change of momentum is equal to the force, i.e., \[ \frac{dp}{dt} = F \] From the definition of momentum, \( p = mv \), where \( m \) is the mass of the electron and \( v \) is its velocity.
Since velocity is proportional to momentum, the rate of change of velocity is related to the force, and thus the de-Broglie wavelength varies inversely with the square root of the distance travelled. Thus, the rate of change of de-Broglie wavelength varies inversely with \( \frac{1}{2} \) power of the distance travelled. Therefore, the value of \( n \) is \( \frac{1}{2} \).