What is the easiest way to create a uniform magnetic field?
Historically, this has been a crucial piece equipment in our cyclotrons, early television sets, mass spectrometers, etc.
This is something you'll likely find covered quite extensively in your undergraduate physics textbook.
Mathematical Derivation
Imagine a current loop in the xy-plane:
dB=4πμ0r2Idl×r^Biosavart Law
Note that this is only for a infinitesimal part of the current loop!
Off the bat, we can make the simplication about the magnitude:
dB=4πμ0r2IdlAlways dl⊥r^
Now we just need to integrate for all the infinitesimal parts of the ring.
To do this, we can realize that the x^,y^ components cancel and only the z^ component will remain.
Hence, we can just add the z^ components.
dBz=4πμ0r2Idlcosθ Just adding the z^ components=4πμ0r2IdlrR=4πμ0(R2+z2)IdlR2+z2R=4πμ0I(R2+z2)3/2Rdl
Note that at the center, the magnetic field is simply:
B(z)=(R2+D2)3/2μ0IR2
Where I is simply total induced current.
How uniform is the hemholtz coil?
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Cyclotron formulation
Electron Gun formulation
Now about dipoles
Now about dropoff at a rate of /r^4
Small experiments
Guitar Coils work in a very similar way but the other way.
Disturbances in the surrounding magnetic field cause a current in the coil that we process as sound.
As a result, when I use my Hemholtz Coil to manually disturb these guitar coils, I can manually change the induced current and sound produced by the guitar.