Question about Faraday's law of induction

In summary, the conversation discusses the Maxwell equation for Faraday's law and how it relates to the electric and magnetic fields. It is noted that the electric field is a conservative vector field, and the equation sets the time derivative of the magnetic field to 0. However, the fact that the electric field is conservative seems to contradict the idea that it is induced by changing magnetic fields. It is explained that this equation relates the production of an electric field to the rate of change of magnetic field intensity, and that electric fields produced in this way are not subject to the restriction of having a curl of 0. There is also a mention of the vector and scalar potentials that contribute to the overall electric field.
  • #1
space-time
218
4
I was studying the Maxwell equation for Faraday's law:

∇×E = -(∂B/∂t)

I then did some math and noticed that the electric field is a conservative vector field, because
∇×E= <0,0,0>

Since this is the case, based on the above Maxwell equation this would set the time derivative of the magnetic field equal to 0 as well (meaning that the magnetic field does not change with respect to time).

If the magnetic field remains constant with respect to time, then what information exactly is supposed to be taken from this equation? Initially, I thought it was supposed to tell you about the electric field that is induced by changing magnetic fields, but the fact that the electric field is conservative seems to disagree with that thought (unless the formula for computing the electric field varies from situation to situation).
 
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  • #2
Hi, space_time. I'm curious about your math to arrive at ∇×E= <0,0,0>.
 
  • #3
Flower_648 said:
Hi, space_time. I'm curious about your math to arrive at ∇×E= <0,0,0>.

E= (KQ/|r|2) * (r/|r|) where r is the position vector <x,y,z>

Transforming r into Cartesian coordinates, this equation turns into:

E= (KQ/(x2 + y2 + z2)^(3/2)) * <x,y,z>

Taking the curl of this yields <0,0,0>.

Do you think I made some kind of arithmetic mistake?
 
  • #4
space-time said:
Do you think I made some kind of arithmetic mistake?
No, looks fine. The definition of the electric field vector field you began with is valid only for electrostatics, though.

It's more generally true that the flux through a surface relates to the enclosed charge, which allows for the field to curl.
 
  • #5
space-time said:
If the magnetic field remains constant with respect to time, then what information exactly is supposed to be taken from this equation? Initially, I thought it was supposed to tell you about the electric field that is induced by changing magnetic fields, but the fact that the electric field is conservative seems to disagree with that thought (unless the formula for computing the electric field varies from situation to situation).

This equation relates the production of an electric field to the rate of change of magnetic field intensity. E produced by a changing B is not conservative. If the curl E is found to be zero you know it is not due to a changing magnetic field intensity since for a conservative field div E =0.

in general E = - ∂A/∂t - V where A is the vector potential due to current densities and V is the scalar potential due to static charges.
 
  • #6
Just to emphasise what's already been said…

Up to the 1830's it was thought that the electric field was conservative (curl E zero everywhere, as we say now), essentially because E was thought to be the sum of fields due to point charges obeying Coulomb's law.

In the 1830's, Faraday (and, I think, Henry) discovered that a magnet thrust into a stationary coil induced a voltage. Since the electrons in the coil weren't moving initially (at least not in a co-ordinated way) it must be an electric field (not a magnetic field per se) that urged them, producing the voltage. This is is the phenomenon summed up by curl E = -dB/dt. [Sorry about lack of partials.]

Electric fields arising in this way (when magnetic fields change) are not subject to the restriction curl E = 0.
 
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Likes Greg Bernhardt and just dani ok

1. How does Faraday's law of induction work?

Faraday's law of induction states that a changing magnetic field will induce an electromotive force (EMF) in a conductor. This EMF can then cause a current to flow in the conductor.

2. What is the significance of Faraday's law of induction?

Faraday's law of induction is important because it explains the relationship between electricity and magnetism. It also forms the basis for many technologies, such as generators and transformers.

3. Can you give an example of Faraday's law of induction in action?

One example of Faraday's law of induction is the way generators use a rotating magnetic field to produce electricity. As the magnetic field changes, it induces an EMF in the wire coils of the generator, creating an electric current.

4. What factors affect the strength of the induced current in Faraday's law?

The strength of the induced current in Faraday's law depends on the rate of change of the magnetic field, the number of turns in the conductor, and the resistance of the conductor.

5. Is Faraday's law of induction related to Lenz's law?

Yes, Faraday's law of induction and Lenz's law are closely related. Lenz's law states that the direction of the induced current will be such that it opposes the change that produced it. This is consistent with Faraday's law, which states that the induced current will flow in a direction to oppose the change in the magnetic field.

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