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Consider a loop made off with a conducting material with resistance $R$. This loop is in a uniform magnetic field. Then, thanks to Faraday's law, we know that a current will be induced in the loop. My question is, what is the nature of the induced current, is it stationary or not? I am asking this question because I would like to compute the force that is exerted in this loop. I know that the force is given by:

$$\vec{F}=q(\vec{v} \times \vec{B}+\vec{E})$$

If the induced current is not stationary, no electric field will be produced, then, $\vec{F}=q(\vec{v} \times \vec{B})$, with $\vec{v}$ the velocity of the loop. But if it is stationary, then, an electric field will be generated... The loop has a $0$ self-inductance.

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  • $\begingroup$ What do you mean by stationary? By definition current is charge/time through a cross-section. The current induced is the result of moving electrons. Did this answer your question or did i not understand it ? $\endgroup$
    – DakkVader
    Commented Jan 11, 2020 at 12:17
  • $\begingroup$ by stationary, I mean that $I=I(t)$ $\endgroup$
    – Dicordi
    Commented Jan 11, 2020 at 12:20
  • $\begingroup$ Well that condition for the current to be induced at all is that the magnetic field ha got to be changing with time, and this means that the current would change with time as well. $\endgroup$
    – DakkVader
    Commented Jan 11, 2020 at 12:22

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The condition for a current being induced in such a loop is that the magnetic flux changes with time. If the loop is fixed and is submitted to a constant magnetic field, there will be no induced current. Look at Faraday's Law:

$$ \oint E \cdot dl = - \frac{\partial \phi_{m}}{\partial t} \; , \phi_{m} = \vec{B} \cdot \vec{A}$$

If neither the magnitude of the magnetic field, nor the magnitude of the area, nor the angle between the magnetic field vector and the area vector, changes, there won't be an induced voltage and hence there won't be an induced current.

Furthermore, even though you change the magnetic flux, creating such an induced current, the value of the current will not be constant. As soon as you stop changing the magnetic flux, the induced voltage will go to 0 and hence the induced current as well. The induced Electric Field is said to be non-electrostatic, which in our context means that there won't be a constant current.

Finally, the Force equation that you have written is only valid for point charges. The magnetic force given by $F_{m} = q(\vec{v} \times \vec{B})$ is associated with moving point charges. Extense bodies such as a loop, a rod, a big wire, etc, have magnetic forces that are calculated differently.

"I agree with the fact that the equation that I gave for the force is only for a point charge, but it was to illustrate my problem, which is: in my situation, how do I compute the force on the loop, is there any electric field that I need to worry about? I forget to say that the loop has a null self inductance."

enter image description here

As you can see in the image, you are able to split your loop into segments such as you can treat these segments as current-carrying wires, and use the magnetic force formula for this case that is: $F_{m} = I(\vec{L} \times \vec{B})$. Use the right-hand rule to find the correct directions and perform a vector sum.

If you change the magnetic flux and create an induced non-electrostatic electric field that will generate an induced current, you will need to take into account that this induced current, by Lenz's law, will be in such a way that it will create a contrary magnetic field that will act opposed to the magnetic field that the loop is submitted. That may change the net magnetic field and therefore the net magnetic force.

Hope I have answered your question, have a nice day!

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  • $\begingroup$ I agree with the fact that the equation that I gave for the force is only for point charge, but it was to illustrate my problem, which is : in my situation, how do I compute the force on the loop, is there any electric field that I need to worry about ? I forget to say that the loop has a null self inductance. $\endgroup$
    – Dicordi
    Commented Jan 11, 2020 at 12:22
  • $\begingroup$ Please see the edits that I have done in my answer. $\endgroup$
    – Victor Lins
    Commented Jan 11, 2020 at 12:34
  • $\begingroup$ @vitor Lins. If I understand this well, the current induced is not constant in time, so, it does NOT produce any magnetic field ? Is that right ? $\endgroup$
    – Dicordi
    Commented Jan 11, 2020 at 16:31
  • $\begingroup$ The induced current is not constant in time, yes! Although it does produce a magnetic field, in fact, by Lenz's law, its direction is opposed to the magnetic field that is in the space involved. The detail is that as soon as the induced current goes to 0 -- when the magnetic flux stops varying -- there will not be any more opposite magnetic field created by the loop. $\endgroup$
    – Victor Lins
    Commented Jan 11, 2020 at 16:34
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$\vec F = \vec q(\vec E + \vec v X \vec B) $ (Lorentz Formula)

gives force on a charge(q) in a electromagnetic field and not force on a loop.

The emf induced in a current loop is given by $ε= -\frac{dΦ}{dt}$ where Φ is the flux of magnetic field and then the current would be $i=\frac{ε}{R}$

As for the force on the loop, a force will be exerted even if the loop is stationary, you can calculate the force by first finding the magnetic dipole formed by the loop.

M= i* (area of the loop) and $F = M\frac{dB}{dz}$ (assuming the loop is in xy plane) now if $\frac{dB}{dz} = 0$, i.e, magnetic field doesn't vary with space, the for on loop shall be zero.

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