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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