HELP - electron diffraction/relativity problem

[hollandrogre]

Homework Statement

A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit. The diffracted beam shows its first diffraction minima at $$\pm$$11.5 degrees from the original direction of the beam when viewed far from the slit. a) Do we need relativity formulas? How do you know? b) How wide is the slit?

Homework Equations

1) M = $$\gamma$$m (possibly needed)
2) $$\lambda$$ = $$\frac{h}{\sqrt{2meV_{accelerating}}}$$
3) $$\theta$$ = $$\frac{\lambda}{w_{slit}}$$

The Attempt at a Solution

I can solve b) by just plugging in my numbers. The problem is a). If the electron is accelerated through 0.1 kV, it will have energy 0.1 keV, which puts its speed at a decent fraction of the speed of light. Classically, we would need Einstein's special relativity equations. However, electron diffraction is a quantum phenomenon - does this make the electron's mass immune to the effects of relativity or do I have to calculate its speed, find the new mass, and use that instead of the rest mass?

 

[anathema]

a) It is important to take into account the relativistic effects on the electron's mass when calculating its speed, as it is traveling at a significant fraction of the speed of light. The correct equation to use for the mass of the electron is the relativistic mass, given by M = \gammam, where \gamma is the Lorentz factor. Therefore, to accurately calculate the electron's speed and resulting diffraction angle, the relativity formulas are necessary.

b) To determine the width of the slit, we can use the diffraction equation \theta = \frac{\lambda}{w_{slit}} and rearrange it to solve for the slit width w_{slit} = \frac{\lambda}{\theta}. Plugging in the values for the wavelength and diffraction angle, we can calculate the width of the slit.

 

[Moetasim]

a) In this scenario, it is necessary to use the equations of special relativity to accurately calculate the speed and mass of the electron. This is because the electron is accelerated to a high enough energy that its speed becomes a significant fraction of the speed of light. In this case, the electron's mass will increase due to relativistic effects, and this must be taken into account when calculating its wavelength and diffraction angle.

b) To calculate the width of the slit, we can use the diffraction equation \theta = \frac{\lambda}{w_{slit}} where \theta is the diffraction angle, \lambda is the wavelength of the electron, and w_{slit} is the width of the slit. From the given information, we can calculate the wavelength using \lambda = \frac{h}{\sqrt{2meV_{accelerating}}}, where h is Planck's constant, m is the mass of the electron, and V_{accelerating} is the potential difference through which the electron was accelerated. By rearranging the equation, we can solve for w_{slit}, giving us the width of the slit.