Waveform of Classic Electromagnetic Induction

[Charles Link]

Very good @alan123hk :)

The peaks in the EMF are closer than expected to the peaks in the flux, (at which point we get a zero in the EMF), but in hindsight, that isn't too surprising.

What is your ratio of $L/d$, where $L$ is the length of the magnet, and $d$ the distance to the small coil?

 

[Charles Link]

@alan123hk From what I derived, I think an identical expression, (the small print is hard to read), you used $L/2=5$, and $d=20$.

 

[alan123hk]

@alan123hk From what I derived, I think an identical expression, (the small print is hard to read), you used L/2=5, and d=20.

Yes, there are two magnetic poles (N and S), the distance between them is 10, and the distance from the one-turn coil to the center of the two magnetic poles is 20.

Adjusting these distance-related parameters will change the induced EMF waveform.

 

[hutchphd]

I have done an analytical model and obtained results consistent with my previous posts (and the lovely experiments).. The model consists if a N and a S magnetic monopole on the ends of a rotating stick of length 2d. A small sensing coil is placed outside the radius of rotation. By varying d (while keeping the dipole moment fixed) one easily reproduces the cos for small d and then flatter then double hump for d approaching the coil. There are no surprises in the model.
Sorry for the tease but I will slog through the LaTeX in the next few days. Too much sh*t on my fan right now!
EDIT: I meant that literally I need to fix my fridge!

We define the dipole using two monopoles on a rotating stick of length 2d with positions $$\mathbf r_N=\binom {d\cos (\omega t )}{d\sin(\omega t)},~~~~~~~\mathbf r_S=-\mathbf r_N $$ Pole "magnetic charges" are assigned as to keep a constant dipole moment $q_M=p/2d$ and the magnetic field from a pole M is then $$\mathbf B(\mathbf r)_M = \frac {\mu_0 q_M } {4\pi}\frac { (\mathbf r-\mathbf {r_M})} { |\mathbf r-\mathbf {r_M}|^3}$$ Our small sensing coil is at position $$\mathbf r_{sense}=\binom c 0 $$where c>d and the coil surface normal is $\hat {\mathbf x}$. The flux through the small coil will then be proportional to the x component of B $$\mathbf B(\mathbf r)_M = \frac {\mu_0 q_M } {4\pi}\frac { (\mathbf r-\mathbf {r_M})} { |\mathbf r-\mathbf {r_M}|^3}$$

Sorry I didn't get this finished. The result comports with @alan123hk exactly and I attach the EXCEL graphs for several values of d/c (note c is not the speed of light...=distance to coil) for a fixed dipole moment. If anyone wants more I will provide same...no sense duplicating effort.

 

[Charles Link]

@hutchphd For your first graph, I believe you have a ratio of $.1$ instead of $.0001$.

Edit: My mistake: I didn't read your explanation that you maintain a constant dipole moment by adjusting the magnetic charge.

 

[hutchphd]

@hutchphd For your first graph, I believe you have a ratio of $.1$ instead of $.0001$.

Edit: My mistake: I didn't read your explanation that you maintain a constant dipole moment by adjusting the magnetic charge.

This is pretty rough I realize...the arduous part is making it pretty. Happy to supply whatever is helpful. Looks like the same result. Incidently you only need define the field for one pole because the second pole is just $\omega t=\pi$ offset with opposite "charge"...makes computation easier.

 

[Charles Link]

The calculation of the magnetic field from a uniformly magnetized cylindrical magnet that we are doing here is one that I think every physics and EE major should be able to do. (See also https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/).

I have seen a question involving this calculation appear in the Physics Forums homework section only on rare occasion=perhaps it, along with E&M in general, needs to see more emphasis in the curriculum. In any case, this is an opportunity for the students to try the calculation, and see if they can come up with the same results that we did.

 

[vanhees71]

We define the dipole using two monopoles on a rotating stick of length 2d with positions $$\mathbf r_N=\binom {d\cos (\omega t )}{d\sin(\omega t)},~~~~~~~\mathbf r_S=-\mathbf r_N $$ Pole "magnetic charges" are assigned as to keep a constant dipole moment $q_M=p/2d$ and the magnetic field from a pole M is then $$\mathbf B(\mathbf r)_M = \frac {\mu_0 q_M } {4\pi}\frac { (\mathbf r-\mathbf {r_M})} { |\mathbf r-\mathbf {r_M}|^3}$$ Our small sensing coil is at position $$\mathbf r_{sense}=\binom c 0 $$where c>d and the coil surface normal is $\hat {\mathbf x}$. The flux through the small coil will then be proportional to the x component of B $$\mathbf B(\mathbf r)_M = \frac {\mu_0 q_M } {4\pi}\frac { (\mathbf r-\mathbf {r_M})} { |\mathbf r-\mathbf {r_M}|^3}$$

Sorry I didn't get this finished. The result comports with @alan123hk exactly and I attach the EXCEL graphs for several values of d/c (note c is not the speed of light...=distance to coil) for a fixed dipole moment. If anyone wants more I will provide same...no sense duplicating effort.

Note that this is the quasistatic approximation though!

 

[hutchphd]

How fast is the bar magnet going to be rotating? I guess it should have been said explicitly.

 

[Charles Link]

How fast is the bar magnet going to be rotating? I guess it should have been said explicitly.

I mentioned that the calculation is a static type in the last paragraph of post 205. That makes it so that it really should be fairly routine, but, I have to wonder whether the majority of the physics and EE students would know how to do the calculation of computing the magnetic field from a uniformly magnetized cylindrical magnet. See also post 217.

 

[Charles Link]

Just an additional comment or two on the above for the calculation of the magnetic field for a cylindrical magnet: Recent conversations with a physics professor who teaches E&M said that these days they are teaching the magnetic surface current method with Biot-Savart to the undergraduate students, (e.g. Griffith's textbook has a problem with a permanent magnet using magnetic surface currents), and they normally don't teach the magnetic pole method until graduate school. For this problem, the magnetic pole method of calculation is the much simpler one, but both methods will get the exact same answer for the magnetic field $B$. If anyone is interested, both methods are outlined in the "link" in post 217.

 

[hutchphd]

As I recall teaching from the first edition of Griffiths in the mid 80's it was done using surface currents. Actually I don't know that I was taught pole strength explicitly in the 70's, but I recalled Dirac's invocation of a monopole as end of a long solenoid.

 

[Charles Link]

My classmates and I were taught the pole method during the late 70's, and we were only shown the surface currents very briefly as an alternative theory. I don't know of any textbook that covers both methods and shows them to be equivalent. The pole method relies on the formula $B=\mu_o H +M$, which is often presented without proof.

See posts 14-16 of https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/. It may be of interest that the pole method formula $B=\mu_o H+M$ can be proven for a system with arbitrary magnetization $M$ starting with Biot-Savart and magnetic current density $J_m=\nabla \times M/\mu_o$, and magnetic surface current per unit length $K_m=M \times \hat{n}/\mu_o$. IMO, this is a much better justification for this formula then simply saying it is analogous to the electrostatic $D=\epsilon_o E+P$, as some textbooks do.

The above proof is done for a system without any currents in conductors. The currents in conductors are then introduced with $H$ being re-defined (besides the magnetic pole contribution), to include a Biot-Savart type contribution to $H$ for currents in conductors. This way, $B=\mu_o H +M$, which is first shown to hold for a system without any currents in conductors, will then still hold when currents in conductors are introduced. This additional $H$ from the conductors, in the form of $\mu_o H$, simply gets added to both sides of the formula $B=\mu_o H+M$, because currents in conductors are always sources for $B$, computed from Biot-Savart.