Electric Field Induced by Varying Magnetic Field

In summary, a uniform magnetic field normal to the plane of the screen with varying magnitude will induce an electric field with concentric circular field lines centered around a point P on the screen. This direction of the electric field is tangent to the circle. This also occurs for a point Q, resulting in two directions of electric field at all points. If we place a stationary electron in this varying magnetic field without any circuit, it will move along a circular path. This is due to Faraday's law, which states that a changing magnetic field produces an electric field. This electric field can influence free charges such as electrons, even without a conducting ring or sheet. However, the characteristics of the magnetic field and its geometry, as well as the location of
  • #36
kcdodd said:
Phrak: When someone says the EM wave equation, this is what I think of. Not of E and B, but of A.

[tex]\nabla^2\phi - \partial^2_t\phi = -\rho[/tex]
[tex]\nabla^2\vec{A} - \partial^2_t\vec{A} = -\vec{J}[/tex]

I could be horribly mistaken, but though these are certainly wave equations, they are not the wave equations of light.

For instance, the homogeneous solution of the wave equation of E is,

[tex]\Box \textbf{E} = 0 \ .[/tex]

[tex] \textbf{E} = - \nabla \phi - \partial _t \textbf{A}[/tex]

[tex]\Box (\nabla \phi + \partial _t \textbf{A}) = 0[/tex]

The last equation is certainly third order in all terms.

It's interesting to note the authors of electromagnetic wave equation on Wikipedia agree with you.
 
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  • #37
I see your point. This is straight out of Jackson, and he seems to side-step the peculiarity of the inhomogeneous wave equations for E and B. (I changed to natural units)

[tex]\Box \vec{E} = \nabla \rho + \partial_t J[/tex]
[tex]\Box \vec{B} = - \nabla \times \vec{J}[/tex]

Which he states has the solution of Jefimenko's equations. So I suppose you are correct that A is implicitly third order here. Which I suppose is ok since the sources are now first order. I'm sure there is some deep meaning here, like the derivative of a wave is another wave but it is getting late and I can't recall a specific relation.
 
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  • #38
kcdodd, you've still brought up an interesting equation, none the less. At one time I was catalouging Lorentz invariant (with other nice properties) equations and gauge invariances of a vector field imposed on spacetime. Somehow I seem to have missed the wave equation for a generalization of the potential, A=(A,phi), or it's not Lorentz invariant (nor connection free). So, thanks for that.
 
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  • #39
To my understanding, the Lorentz gauge condition is Lorentz invariant (but not many other gauge conditions, e.g. the Coulomb gauge condition is not Lorentz invariant). I don't know about the connection. It would be nice to know if the Lorentz gauge is unique in being Lorentz invariant, or if it is one of a whole class of gauge conditions which are Lorentz invariant, but that is beyond my knowledge currently.
 
  • #40
Dale, my apologies for being misleading. I know remarkably little about electromagnetism in vector calculus and gauge fixing--or even electromagnetism in mixed tensor notation. As far as I can tell, everything in electromagnetism is expressible as antisymmetric covariant tensors in its native four dimensions. Equations written in these are by design, Lorentz invariant and connection free. So this is what I've studied instead of the tools of vector calculus.

Every tensor field that is over scalar-valued has a regauging field of one less index--and it may be possible to bend the rule on scalars, come to think of it. The charge-current density for instance, can be expressed as either a covariant vector or a tensor of three lower indices.
The last is regaugable by an electromagnetic field tensor.

In generalizing the covariant 4-vector potential to complex entries, an interesting effects on regauging occurs; the imaginary part of the vector potential regauges the electromagnetic field tensor, and the real part regauges it's dual, Gmu nu = (1/2)epsilonmu nurho sigmaFrho sigma. Maxwell's equations obtain their maximum symmetry in a single expression, magnetic monopoles appear, and now the problem is getting rid of them.

A scattering of other symmetries and conservation laws come up as well whether using real or complex fields, all beginning with the simple notion of applying a 4-vector field to spacetime.

Sorry for all the greek. The first two chapters of Sean Carroll's Notes on General Relativity provide the background for all this. As much as the 4-velocity provides the unification of mass, energy and momentum, the covariant 4-vector potential does the same for electromagnetism.
 
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  • #41
Phrak said:
As much as the 4-velocity provides the unification of mass, energy and momentum, the covariant 4-vector potential does the same for electromagnetism.
Hmm, I should look into that more. It sounds like something that I would enjoy.
 
  • #42
kcdodd said:
Phrak: When someone says the EM wave equation, this is what I think of. Not of E and B, but of A.

[tex]\nabla^2\phi - \partial^2_t\phi = -\rho[/tex]
[tex]\nabla^2\vec{A} - \partial^2_t\vec{A} = -\vec{J}[/tex]

Maxwell: -"To find the rate of propagation of transverse vibrations through the elastic medium, on the supposition that its elasticity is due entirely to forces acting between pairs of particles

[tex]V=\sqrt{m\over p}[/tex]

where 'm' is the coefficient of transverse elasticity, and 'p' is the density."

The "wave equation" you are talking about is nothing else but the 'wave equation for vibrating string': - "The speed of propagation of a wave in a string (v) is proportional to the square root of the tension of the string (T) and inversely proportional to the square root of the linear mass (μ) of the string:

[tex]\frac{\partial^2 y}{\partial x^2}=\frac{\mu}{T}\frac{\partial^2 y}{\partial t^2} => v = \sqrt{T \over \mu}[/tex]

".
- http://en.wikipedia.org/wiki/Vibrating_string
 
  • #43
DaleSpam said:
Hmm, I should look into that more. It sounds like something that I would enjoy.

I hope you do. I don't know anyone conversant in this.
 
<h2>1. What is the concept of electric field induced by varying magnetic field?</h2><p>The concept of electric field induced by varying magnetic field is based on Faraday's law of electromagnetic induction, which states that a changing magnetic field can induce an electric field. This means that when a magnetic field is varied or moving, it can create an electric field in the surrounding space.</p><h2>2. How is the electric field induced by varying magnetic field measured?</h2><p>The electric field induced by varying magnetic field can be measured using a device called a magnetometer. This instrument can detect changes in the magnetic field and convert it into an electric signal, which can then be used to calculate the strength and direction of the electric field.</p><h2>3. What are some practical applications of electric field induced by varying magnetic field?</h2><p>The electric field induced by varying magnetic field has many practical applications, including power generation in electric generators, wireless charging of electronic devices, and magnetic resonance imaging (MRI) in medical diagnostics. It is also used in various industrial processes such as metal processing and particle accelerators.</p><h2>4. How does the strength of the electric field induced by varying magnetic field depend on the strength of the magnetic field?</h2><p>The strength of the electric field induced by varying magnetic field is directly proportional to the strength of the magnetic field. This means that the stronger the magnetic field, the stronger the induced electric field will be. However, the rate of change of the magnetic field also plays a role in determining the strength of the induced electric field.</p><h2>5. Can the direction of the electric field induced by varying magnetic field be controlled?</h2><p>Yes, the direction of the electric field induced by varying magnetic field can be controlled by changing the direction of the magnetic field or by adjusting the rate of change of the magnetic field. This is the principle behind devices such as electric motors, where the direction of the electric field is manipulated to produce motion.</p>

1. What is the concept of electric field induced by varying magnetic field?

The concept of electric field induced by varying magnetic field is based on Faraday's law of electromagnetic induction, which states that a changing magnetic field can induce an electric field. This means that when a magnetic field is varied or moving, it can create an electric field in the surrounding space.

2. How is the electric field induced by varying magnetic field measured?

The electric field induced by varying magnetic field can be measured using a device called a magnetometer. This instrument can detect changes in the magnetic field and convert it into an electric signal, which can then be used to calculate the strength and direction of the electric field.

3. What are some practical applications of electric field induced by varying magnetic field?

The electric field induced by varying magnetic field has many practical applications, including power generation in electric generators, wireless charging of electronic devices, and magnetic resonance imaging (MRI) in medical diagnostics. It is also used in various industrial processes such as metal processing and particle accelerators.

4. How does the strength of the electric field induced by varying magnetic field depend on the strength of the magnetic field?

The strength of the electric field induced by varying magnetic field is directly proportional to the strength of the magnetic field. This means that the stronger the magnetic field, the stronger the induced electric field will be. However, the rate of change of the magnetic field also plays a role in determining the strength of the induced electric field.

5. Can the direction of the electric field induced by varying magnetic field be controlled?

Yes, the direction of the electric field induced by varying magnetic field can be controlled by changing the direction of the magnetic field or by adjusting the rate of change of the magnetic field. This is the principle behind devices such as electric motors, where the direction of the electric field is manipulated to produce motion.

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