Potential Difference

To determine the factor by which the de Broglie wavelength of an electron changes when the potential difference used to accelerate it is quadrupled, we start with the relevant formulae. The de Broglie wavelength λ is given by:

$ is Planck's constant and is the momentum of the electron.

The momentum can be expressed in terms of the kinetic energy as:

\) acquired by the electron is due to the potential difference , so , where is the charge of the electron.

Thus, the momentum becomes:

\) into the de Broglie wavelength formula:

\) is quadrupled, the new wavelength when the potential is is:

\) with the original :

\) $

This means the de Broglie wavelength is halved. The factor by which the wavelength changes is 0.5.

An electron is accelerated from rest through a potential difference of 200 V. To find the de Broglie wavelength associated with this electron, we apply the following steps:

1. Kinetic Energy (KE): When an electron is accelerated through a potential difference , it gains kinetic energy equal to the work done by the electric field:
, where .

2. Calculating KE: Here, , so:
.

3. Relating KE to momentum: The kinetic energy of the electron also relates to its momentum by the formula:
, where .

4. Solving for momentum (p): By rearranging, :


.

5. Applying de Broglie wavelength formula: The de Broglie wavelength is given by:
, where (Planck's constant).

6. Calculating wavelength:
.

7. Converting meters to nanometers: , so:

8. Rounding: The wavelength is to 2 decimal places.

Verification: The calculated de Broglie wavelength .