Derivation of the Bohr Magneton

[aeroegnr]

This is a homework problem, and I already turned in the wrong answer (on purpose because I didn't agree with the explanation of why the correct answer was twice mine). I want to know why the answer is what it is. The stated book value is 9.274009 x 10^-24 J/T. I got exactly half that, and I know the equations that the official solution used.

The question is stated thus:
a) the current i due to a charge q moving in a circle with frequency f_rev is q*f_rev. Find the current due to the electron in the first bohr orbit.


So, what I did was I used the equation h*f=E, where E was the energy of the first orbit in hydrogen, which was 13.6eV. (I know that this is where I made the mistake) I then computed the current that way and got 9.274.../2 as the answer. The book solution manual, which I do not trust because it offers no explanation, used this equation:

f~Z^2*m*k^2*e^4/(2*pi*h_bar^3*n^3) and plugged in the value of n=1 to get the frequency.

however, the above equation is an approximation for large n. The actual equation that the above is derived from is:

Z^2*m*k^2*e^4/(4*pi*h_bar^3) * (2n-1)/(n^2*(n-1)^2)

I know that for large n, this equation approaches the other one they used. However, they plugged in the value of 1 into the approximation, when the real answer would have been undefined (divide by 0)!

I was told by the professor that I could use the equation f=v/(2pi*r), which did not suit me because you end up with the approximation equation above.

What am I confused about here?

 

[theFuture]

Well, if the book is using the approximation and you are only off by a factor of two my guess is that you are fine. You work seems fine.

 

[R0man]

The Bohr magneton is a fundamental constant in quantum mechanics that describes the magnetic moment of an electron in an atom. It is given by the formula μB = eh/4πm, where e is the charge of an electron, h is Planck's constant, and m is the mass of an electron.

In order to derive this, we need to consider the motion of an electron in the first Bohr orbit. The electron is moving in a circular path with a frequency f_rev, which is given by the equation f_rev = q*f, where q is the charge of the electron and f is the frequency of the electron's motion. In this case, q is the same as the charge of an electron, which is -e.

The correct approach to finding the current due to the electron in the first Bohr orbit is to use the equation I = q*f_rev. In this case, q is -e and f_rev is the frequency of the electron's motion in the first Bohr orbit. This frequency can be found using the equation f_rev = v/(2π*r), where v is the velocity of the electron and r is the radius of the orbit. In the first Bohr orbit, the velocity of the electron can be found using the equation v = k*e^2/(2r), where k is a constant.

Putting these equations together, we get I = -e*(k*e^2/(2r))/(2π*r) = -ke^3/(4πr^2). This is the current due to the electron in the first Bohr orbit. To find the magnetic moment, we use the formula μ = I*A, where A is the area of the orbit. In this case, A = πr^2, so μ = -ke^3/(4πr).

Now, we can use the equation for the energy of the first Bohr orbit, E = -k*e^2/(2r), and rearrange it to solve for r. This gives us r = -ke^2/(2E). Plugging this into our equation for the magnetic moment, we get μ = -ke^3/(4π*(-ke^2/(2E))^2) = -ke^3/(4π*(-k^2*e^4/(4E^2))) = -ke^3/(4π*(-k^2*e^4/4*13.6^2))