Question

So I've been banging my head against this problem for a while and I genuinely have no clue how my answer is differing than the one provided to me and while I'm not usually the type to ask such a thing, I want to make sure I'm not missing anything. So here's what I did:

First thing's first, a state of 3d gives us n=3,andl=2
This is nice since it's kinda all we have/should need. We also know that μ=I⋅A
where I
is the current in amps, A
is the area of the circle that the orbit of the electron creates and μ
is our magnetic moment. Cool, we also know that μ=μ√(Bl(l+1))
(μB is the Bhor Magneton, around 9.3⋅10−24J/T) and because we have a hydrogen atom we also have that our radius can be found with rn=n²a₀ where n is our principal number and a is the base radius of hydrogen (something like 0.529165899 angstroms). This gives us a radius of 9a₀

So we mash all these together and plug in to find I right?

I=μ/A=μ√(Bl(l+1))/πr²=μB√(2(2+1))(π⋅(3²⋅a₀)²)= 3.18801⋅10−5A

Except, it seems the answer is NOT this, instead, it's apparently supposed to be 4.43⋅10−7A. So either the book is wrong (seeming more and more plausible) or I'm wrong, and I really want to know where I screwed up. I AM NOT ASKING FOR THE SOLUTION. What's the point of learning then? All I need to know is what step I've missed so I can put the puzzle together myself.

Thanks so much anyone who comes to my aid.

Edit: I've realized that I should make people aware that we are kinda just assuming that the Bohr model of perfect circles is true or at least true enough for the electron to mimic a current in a wire loop while orbiting the nucleus. It's not true and I'm aware, this idea is more for an introduction in the electromagnetic mechanics that go on with atoms (really just hydrogen atoms). The main thing this was for was to introduce a very general idea of the Zeeman effect and similar phenomenon. We are also ignoring spin here since this question is being asked in the chapter before it was explained so I'm assuming I'm not supposed to worry about it.

Answer 1

This question deals with the intricate calculation of the magnetic moment and associated current of an electron in the 3d state of a hydrogen atom using Bohr's model and quantum mechanics principles.

Step-by-step explanation:

The core of this question revolves around examining the magnetic moment (μ) of an electron in a specific atomic state, more specifically, the 3d state of a hydrogen atom, and why the calculated current (I) associated with this state is different from the expected value. The attempt to find the correct current using the formulas for magnetic moment, Bohr magneton, orbital radii in the Bohr model, and the quantum mechanical definition of angular momentum lays out the intricate connection between quantum numbers, electron orbits, and magnetic effects. The detailed calculation involves understanding the Bohr model, the angular momentum quantum number (l), and the radius of the electron's orbit (rn), and combining these with Planck's constant (h) to elucidate the magnetic properties of an atomic electron.