# Force on a Magnet in Lenz's Law

## Homework Statement

As the bar magnet (see attached diagram) leaves the loop of wire, it experiences a force opposing its exit. I understand why, according to Lenz's law, a force would oppose the bar magnet's exit. I don't understand, however, what causes the force. In this case, the force is rightwards.

In other words, what are the mechanics that cause the magnet to experience a rightwards force?

## Homework Equations

See diagram.

Lenz's Law: Current is induced in a surface to oppose the change in magnetic flux through it.

## The Attempt at a Solution

Here, the current induced produces a magnetic field that points in the same direction as the magnet. Hence, as the magnet tries to leave, another magnetic field somehow opposes its motion...

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## Answers and Replies

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andrevdh
Homework Helper
The moving magnet causes a change in magnetic flux through the coil. This induces current in the coil according to Faraday' s law of electromagnetic induction. The current in the coil now sets up a magnetic field of its own. Lenz's law helps us determine in which direction the induced current in the coil will be.

Hesch
Gold Member
In other words, what are the mechanics that cause the magnet to experience a rightwards force?
In short: Lenzes law says that the nature will counteract any changes in magnetic flux. So when pulling the magnet away from the loop, a current will be induced in the loop that substitutes the flux of the magnet by its own flux. All in all the amount (volume) of flux is increased.

The density of energy in a magnetic field, Emagn = ½*B*H [ J/m3]. So when magnet and loop are separated, energy must be created, to fill out the "added volume" of magnetic energy density.

F = ΔE / Δs = ( Emagn * ΔV ) / Δs = Emagn * A. ( s is distance, A is cross section area of field, V is volume of field ).

andrevdh
Homework Helper
All that Lenz's law enables us to say is that in this case the induced current will increase the flux through the coil and thereby opposing the removal of the magnet. It is just a qualitative law not quantitative. You seem to try and evaluate the opposing force that the magnet is experiencing from the induced magnetic field.

Hesch
Gold Member
You seem to try and evaluate the opposing force that the magnet is experiencing from the induced magnetic field.
I'm finding the opposing force by means of energy in a magnetic field, and are not using Lenz's law to do this.

andrevdh
Homework Helper
Don't think I can help you there, but keep in mind that it is the change in magnetic flux that brings this about. That is the induced current causes an increase in flux, not necessarily such that the total flux is kept constant. In practice one would probably measure the required force to remove the magnet and thereby calculate the induced flux change responsible for it.

Hesch
Gold Member
That is the induced current causes an increase in flux
The flux is not increased, the induced current in the loop will try to keep flux steady, and could do so, were the loop a superconductor. But the volume of the fluxdensity is increased, thus the energy is increased.

What is wrong with this:

F = ΔE / Δs = ( Emagn * ΔV ) / Δs = Emagn * A. ( s is distance, A is cross section area of field, V is volume of field ).
? ?

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andrevdh
Homework Helper
I would have thought that the induced current created more flux lines and thereby increased the overall energy density, but I am not at work now and can't reference my sources.

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andrevdh
Homework Helper
The induced magnetic field from the current in the coil is in the same direction as that of the magnet that is removed
from the coil. The coil thus acts as an additional magnet. It attracks the magnet as it is removed from the coil.
The source of the force is thus the coil. This effect was also used in old car speedometers - Arago's rotation effect
http://www.physics.montana.edu/demonstrations/video/5_electricityandmagnetism/demos/aragosdisc.html
https://prezi.com/dyvhcumzt6k_/electromagnetism/
The two objects become locked together and a force is required to separate them. Hope this is what you wanted to know.

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Hesch
Gold Member
I would have thought that the induced current created more flux lines and thereby increased the overall energy density
This is wrong. The induced current in the loop will just try to keep the flux steady. It will never increase the energy density, but (here) the volume in which the steady energy density acts, thus increasing the total magnetic energy: ΔE = Emagn*Δvolume. The loop-current, creating some flux, will just substitute the disappearing flux from the magnet as it is removed. This leads to:
F = ΔE / Δs = ( Emagn * ΔV ) / Δs = Emagn * A. ( s is distance, A is cross section area of field, V is volume of field ).
I wanted to know if you agreed with that calculation?

Say that the flux were increased through the loop, a back-emf would immediately be created in the loop since emf = dψ/dt.

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andrevdh
Homework Helper
So your formula is suggesting the force is a result of a gradient created in the magnetic energy density?
The term Emagn*ΔV (have for a long time) worries me.
What does this term mean?
What would the ΔV signify?

Hesch
Gold Member
ΔV = change in volume.
Emagn * ΔV = ( magnetic energy density ) * Volume. In units: [ J / m3 ] * [ m3 ] ) = [ J ].

andrevdh
Homework Helper
As you move a distance Δs along the field you go through a small volume ΔV where the field exists?

Hesch
Gold Member
Yes: Δs * A = ΔV. ( A = cross section area ).

andrevdh
Homework Helper
I still would think that you need an energy gradient for a force to be present, that is the energy needs to change if
you moved a small distance in the field, that is Emagn should change.

Hesch
Gold Member
No, the total energy must change:

F = ΔE / Δs = ( Emagn * ΔV ) / Δs = Emagn * A.

andrevdh
Homework Helper
According to some source I can get my hands on the force is proportional to - dH2/ds
but I really don't know enought about the subject.

Hesch
Gold Member
Emagn = ½*B*H = ½*μ0*H*H. Thus the force is proportional to - dH2/ds.

andrevdh
Homework Helper
Sorry, I am not happy with what you are saying.
I would think that the force should be dependent on the change in the energy
and not just its value, that is if the energy stays constant there would be no force.

Hesch
Gold Member
Yes, that's what I'm saying: F = ΔEtotal / Δs.

ΔEtotal = 0 → F = 0

Etotal is not the same as Emagn

andrevdh
Homework Helper
Yes, but you final answer, Emagn*A, do not reflect that (I might be wrong).

Hesch
Gold Member
Again: What is wrong in this?

F = ΔE / Δs = ( Emagn * ΔV ) / Δs = Emagn * A. ( s is distance, A is cross section area of field, V is volume of field ).

ΔV = A * Δs.

andrevdh
Homework Helper
Sorry, I don't know, but physically I just cannot wrap my head around what it is actually saying.
Your introduction of ΔV looks a bit artificial to me.

Hesch
Gold Member
Volume = (length*width)*height = basearea*height

Δvolume = basearea * Δheight

andrevdh
Homework Helper
This ΔE/Δs asks how the energy density is changing when you move around in the space.
Your answer suggest this is how it is changing: Emagn*A ?
That does not seem sensible to me or I am just not understanding your symbols correctly.