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

It is given that the rate of change of the de-Broglie wavelength \( \lambda \) of an electron is inversely proportional to the \( n \)th power of the distance traveled. The objective is to determine the value of \( n \). The de-Broglie wavelength \( \lambda \) of an electron is defined as: \[\lambda = \frac{h}{p}\] where \( h \) represents Planck's constant, and \( p \) denotes the momentum of the electron. The momentum of the electron within the electric field of a sheet is affected by the force exerted on it by the sheet, which is contingent upon the electric field produced by the sheet. The electric field \( E \) generated by a uniformly charged sheet is given by: \[E = \frac{\sigma}{2\epsilon_0}\] The force \( F \) acting on the electron is: \[F = eE = e \frac{\sigma}{2\epsilon_0}\] The rate of change of momentum is equivalent 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. As velocity is proportional to momentum, the rate of change of velocity is related to the force. Consequently, the de-Broglie wavelength varies inversely with the square root of the distance traveled. Thus, the rate of change of de-Broglie wavelength is inversely proportional to the \( \frac{1}{2} \) power of the distance traveled. Therefore, \( n \) is equal to \( \frac{1}{2} \).