How Does Electromagnetic Theory Explain Forces in a Moving Rod?

[Fenter]

This question is a little bit more difficult. I'm not too sure if I'm in the right place to be posting this type of question- but here it is:

Type: Theory Question regarding electromagnetism

My professor is an extremely difficult person to understand. His notes are all over the place and his equations are not labeled or explained in the slightest.

At the end of his lecture, he gave five equations- None of which are labeled.

I will give you a background of what he was lecturing about with equations along the way to help you understand what was being learned (for those who have taken such a course), then I'm going to give you the five equations at the end for deciphering. Please help me! I would deeply appreciate it.

He lectured about the law of biot and Savart (spelled correctly?) by saying that all current loops induce a magnetic field such that B = ((muo)(NiA))/((2)(pi)(r)). He then proceeded to say that the derivative of the flux of a magnetic field with respect to time is the negative of electromagnetic force. He then applied this to crossbars in a circuit- which when moved at an angle to a magnetic field- would induce a current such that the force resisting it was Fb = iL x B. Since emf = d(flux)/dt = d(B*A)/dt (as given by Ampere's Law) if a rod of Length W moves along a u shaped circuit through a distance L, then d(B*A)/dt --> B*W*(dL/dt). (dl/dt) = velocity = v. Thus emf = B*W*v. This professor went on about how if current went through the rod, the rod would move through the magnetic field and gave equations to show the force that would do this. He then proceeded to show how if the rod were stopped by an external force, the rod would heat up (P = i^2R).

There is nothing more to this lecture other than the five seemingly meaningless equations at the end of the lecture.

Here are the five equations (I will use PR for the partial derivative sign)

1. (Gradient) x E = -PR(B)/PR(t)
2. E = -(Gradient)v - PR(A)/PR(t) For PR(A), A = Area of current loop
3. B = (Gradient) x A
4. (Gradient) dot B = 0
5. (Gradient) dot E = Resistivity/(Permitivity of free space)

I also appologise for the lack of vector notation. It should be pretty obvious which is a vector and which is not.

I hope you can help me with these. I appologise for not not making these equations more beautiful. I'm not sure how many other posters were able to make those images of various equations. In time, I'll figure it out. I hope it's not something as simple as reading the FAQ. In any event- this algebra should not be too difficult to read. The idea behind the lecture is there- anybody who understands the fundamentals of what was written will most definitely know what those five equations mean thus majority of the math is not necessary.

Best Regards,

- Fenter

 

[Fenter]

Still Stuck

I'm beginning to wonder if he was just hinting at future lectures. One of these equations has been addressed in a future lecture. HOWEVER, It's not totally addressed and the problem as to the physical meaning of the equation still puzzles me. it's E = -(Gradient)v. What in the world does the partial derivative of each component in a vector with respect to something mean? Isn't that essentially the derivative?

Furthermore, what IS gradient? I know it's an operator that takes the partial derivative of each component in a vector with respect to SOMETHING- but what? Anything? Does it need to be specified? Can anybody help me?

 

[ogb p]

Dear Fenter,

Thank you for reaching out for help with understanding these equations. I can see why you are struggling, as they are not labeled and can be quite confusing.

Firstly, let's go through the background information you provided. Your professor was discussing the law of Biot and Savart, which states that a current loop will induce a magnetic field. This magnetic field can be calculated using the equation B = ((muo)(NiA))/((2)(pi)(r)), where muo is the permeability of free space, N is the number of turns in the loop, i is the current, A is the area of the loop, and r is the distance from the center of the loop to the point where you are measuring the magnetic field.

Your professor then discussed the derivative of the flux of a magnetic field with respect to time, which is equal to the negative of the electromagnetic force. This is known as Faraday's law of induction. He then applied this to a crossbar in a circuit that is moving at an angle to a magnetic field, which would induce a current. The force resisting this motion is given by Fb = iL x B, where i is the current, L is the length of the crossbar, and B is the magnetic field.

Next, your professor discussed how if the crossbar were moving along a u-shaped circuit, the change in flux would be equal to B*W*v, where B is the magnetic field, W is the width of the rod, and v is the velocity of the rod. He then showed how if the rod were stopped by an external force, it would heat up, which is given by the equation P = i^2R, where P is the power, i is the current, and R is the resistance.

Now, let's look at the five equations your professor gave at the end of the lecture:

1. (Gradient) x E = -PR(B)/PR(t)
This equation is known as Maxwell's equation for electromagnetic induction. It relates the electric field (E) to the rate of change of the magnetic field (PR(B)/PR(t)).

2. E = -(Gradient)v - PR(A)/PR(t)
This equation is known as the electromotive force equation. It relates the electric field (E) to the velocity (v) and the rate of change of the area of the current loop (PR(A)/PR(t)).

3. B = (