Going From EMF to Faraday's Law to find E-field in my HW problem

AI Thread Summary
The discussion revolves around understanding the relationship between electromotive force (EMF), Faraday's Law, and electric fields (E-fields) in the context of a toroidal coil problem from Griffiths' textbook. The confusion arises from the constant magnetic flux (Φ) despite a changing current (I), leading to questions about the substitution of variables and the equivalence of E-fields and magnetic fields (B-fields). Participants clarify that if the E-field is purely induced, it follows the same equations as a magnetostatic field, with the current density replaced by the negative rate of change of the magnetic field. The discussion emphasizes the need for a deeper understanding of the mathematical relationships and the application of relevant equations in the problem. Ultimately, the participants aim to reconcile their understanding of these concepts to solve the homework problem effectively.
ChanceLiterature
Messages
9
Reaction score
1
Homework Statement
Ok, so this out of Giffth's intro to Edynamics. Problem 7.19:
A toroidal coil has a rectangular cross section with inner radius a, outer radius a+w and height h. It has N tightly wound loops. dI/dt= k. Additionally, w, h << a. Find E above center of toroid at height z.
Relevant Equations
B inside toroid mu NI/ 2 /PI / s; s being from cylindrical.
B outside = 0
Ok, so I understand how to find dphi/dt that is integral of -d/dt(B "dot" da). In this case I find a Phi that is a constant in space in time which causes me confusion in next step.
Edit: dphi/dt is constant...

Grithff's then says E field same as a Mag field above center of circular current. He writes the B found from solving Biot-Savart for a ring with current I, and says that in this senecio I is equal to -1/mu dPhi/dt.

Firstly, how the heck am I supposed to know to make this substitution based on the textbook?Secondly, how does this make sense. How can I take a statement about EMF and faradays law that will gives me E dot dl of a closed loop and use that to find the E at any point in space. Especially, when the EMF is not a function of time or space!

I understand the idea that E field in this problem is equivalent to the B field in the prior problem. However, I can't explicitly understand how they are the same and I couldn't find any mathematical justification of there equivalence.

I would appreciate any help anyone can give me. This problem has frustrated me to no end. Therefore, I apologize for any moodiness while writing the problem.
 
Last edited:
Physics news on Phys.org
Problem:
1649203144806.png

Solution from solution manual:
1649203225824.png
 
ChanceLiterature said:
Homework Statement:: Ok, so this out of Giffth's intro to Edynamics. Problem 7.19:
A toroidal coil has a rectangular cross section with inner radius a, outer radius a+w and height h. It has N tightly wound loops. dI/dt= k. Additionally, w, h << a. Find E above center of toroid at height z.
Relevant Equations:: B inside toroid mu NI/ 2 /PI / s; s being from cylindrical.
B outside = 0

Ok, so I understand how to find dphi/dt that is integral of -d/dt(B "dot" da). In this case I find a Phi that is a constant in space in time which causes me confusion in next step.
The current ##I## is changing with time. How can the flux be constant?

ChanceLiterature said:
Grithff's then says E field same as a Mag field above center of circular current. He writes the B found from solving Biot-Savart for a ring with current I, and says that in this senecio I is equal to -1/mu dPhi/dt.

Firstly, how the heck am I supposed to know to make this substitution based on the textbook.

Secondly, how does this make sense. How can I take a statement about EMF and faradays law that will gives me E dot dl of a closed loop and use that to find the E at any point in space. Especially, when the EMF is not a function of time or space!

I understand the idea that E field in this problem is equivalent to the B field in the prior problem. However, I can't explicitly understand how they are the same and I couldn't find any mathematical justification of there equivalence.
He explains all of this pretty clearly at the beginning of that section 7.2.2 (in the fourth edition). If ##\mathbf{E}## is a purely induced field, it satisfies the same equations as a magnetostatic field with ##\mu_0 \mathbf{J}## replaced by the ##-\partial_t \mathbf{B}##.
 
  • Like
Likes Delta2 and ChanceLiterature
Apologizes, I meant to say dphi/dt is constant in time and space.
Ok, I will look over that section in more detail now.
 
vela said:
The current ##I## is changing with time. How can the flux be constant?He explains all of this pretty clearly at the beginning of that section 7.2.2 (in the fourth edition). If ##\mathbf{E}## is a purely induced field, it satisfies the same equations as a magnetostatic field with ##\mu_0 \mathbf{J}## replaced by the ##-\partial_t \mathbf{B}##.
Wow! I have no idea how I missed that! That raises a different question though. If we have EQ 7.18 from section 7.2.2, then why in the solutions does Griffiths find phi first when EQ 7.18 only requires partial of B. Do I need to find phi?
 
The ##\partial_t B## is like the current density ##J##, but the magnetic field due to the loop depends on the current ##I##, which is analogous to the magnetic flux ##\Phi_B##.
 
  • Like
Likes ChanceLiterature and Delta2
Thank you!
 
Back
Top