Relativistic Kinetic or Classical?

[MostlyHarmless]

Homework Statement

Suppose you want to make a velocity selector that allows undeflected passage for electrons whose kinetic energy is $5x10^4eV$. The electric field available to you $2x10^5V/m$. What magnetic field will be needed?

Homework Equations

$u=\frac{E}{B}$
u is velocity, E is the electric field, B is the magnetic field.

The Attempt at a Solution

So I'm trying to using the Kinetic Energy to solve for u, and then solve for B.

At first I tried using the classical equation for kinetic energy, which gave me ~.44c. But this value for u, would make the Classical formula not very accurate.

So I tried the relativist formula: $E_{kin}=mc^2(\gamma-1)$ and worked my way down to $u = c\sqrt{1-\frac{1}{\frac{E^2}{(mc^2)^2)}+1}}$

In this case, I end up with ~.01c which is not a relativistic speed. So my question is, which should I use? And obviously I'm doing something wrong, any obvious mistakes with that last equation? Is there an easier way to deduce the magnetic field given the kinetic energy and electric field?

 

[Oxvillian]

and worked my way down to $u = c\sqrt{1-\frac{1}{\frac{E^2}{(mc^2)^2)}+1}}$

I think it's just an algebra error. I got a very slightly different formula at the end

 

[maajdl]

First, I had to understand what a velocity selector is.
Here is a clear picture from Wikipedia:

Indeed, the solution boils down to calculating the velocity of a beam of electron at the enrgy of 5.10⁴eV.
The energy is related to the speed by this formula:

Solving, you should find v/c = 0.412687

v²/c² = (1 - 1 / (Ek/mc² + 1)²)

The classical formula is not so far from the relativistic result.
Anyway, the relativistic formula is more general and is always applicable.
The classical formula is an approximation.

 

[MostlyHarmless]

I think in the frustration I broke an algebra rule, and did: ((E/mc^2)+1)=(E/mc2)^2+1^2

>.<

 

[paranormal]

I would recommend using the relativistic formula for kinetic energy. While the classical formula may give you a quicker solution, it is not as accurate when dealing with high energies and speeds. The relativistic formula takes into account the effects of special relativity, which become more significant at high speeds.

As for your calculation, it seems like there may be an error in your equation for u. The correct equation is $u = c\sqrt{1-\frac{1}{\frac{E_{kin}}{(mc^2)^2}+1}}$, where $E_{kin}$ is the kinetic energy in Joules and $m$ is the rest mass of the electron in kilograms. Also, make sure to convert the kinetic energy from eV to Joules before plugging it into the equation.

To find the magnetic field, you can rearrange the equation $u=\frac{E}{B}$ to solve for B. You should get $B=\frac{E}{u}$. Plugging in the values given in the problem, you should get a magnetic field of approximately 2.0 Gauss.

In summary, as a scientist, I would recommend using the relativistic formula for kinetic energy and double-checking your calculations to ensure accuracy.