Force on a shrinking circular loop in a constant B field

[Coffee_]

1. A conducting loop is shrinking in a constant $\vec{B}$-field that is perpendicular to the loop-axis. The radius of the loop is given by $r(t)=r_{0}e^{-at}$. The resistance in the loop is $R(t)=2 \pi \rho r(t)$. What force is needed to keep the loop shrinking like this?2. Faraday's law of induction, the force on a piece of wire in a constant B field, basic mechanics force/power equations and such.3. I have no trouble finding the functions $\epsilon(t)$, $I(t)$ , $R(t)$.

This means that the power of dissipation through resistance is known as well $P(t)$. Now I have the basis for two reasonings but can't seem to finish neither of them.

The first reasoning is using the fact that we know that a piece of wire $\vec{dl}$ with current $I$ experiences a force in a magnetic field equal to $\vec{dF}=I\vec{dl}\times\vec{B}$. So I know what force each single piece will experience, which seems close to the answer but I seem to not be able to make that step. Integrating this over the loop will give 0 at each moment which is logical since the center of mass isn't moving.

The second reasoning is considering the $P(t)$ dissipated by the resistance in the loop. I'm not sure but I think that this power is exactly the power of the force that is shrinking the loop. I'm not sure however because the current is increasing in time, so beyond dissipation of energy through the resistance another energy sink might be present in the loop? Anyway if we can assume that the dissipated power is the only loss of energy in the loop, I know that for simple compositions one could do something like $\vec{F} \vec{v}=P(t)$ but it's slightly more complicated here.

So I would aprreciate some help.

 

[BiGyElLoWhAt]

Hmm... I don't have a solid reasoning for this, but my thoughts are:
1 nature tries to preserve itself, meaning that the force on the wire should be outwards radially. If you integrate this in cartesian (or just approximate via force field lines) the sum relative to a cartesian coordinate system should be zero, because it's symetrical about any line that cuts the center of the circle. would this be the same in radial coordinates? the field lines would be pointing outward from the center for any theta, which would imply a non-zero force.
2 actually, I'm not really sure if I have a second point. If you can express F in cylindrical coordinates, I think that may fix the problem, I haven't done so yet, but I will attempt to here shortly. My main concern here (i'm not even sure how concerned i should be) is that this problem seems unsolveable in one coordinate system, but perfectly solveable in another. That being said, the net force, as you ointed out, on the ring should be zero relative to any cartesian coordinate, as the center of mass is satationary, while the ring definitely moves with respect to r (its shrinking)

 

[haruspex]

a constant B⃗-field that is perpendicular to the loop-axis

Is this right? The field is parallel to the plane of the loop? I'll assume it means the field is parallel to the axis.

What force is needed to keep the loop shrinking like this?

My guess is that this means the tension in the wire.

 

[BiGyElLoWhAt]

I believe that there is an external force being applied to the loop that causes it to shrink, and that force must counteract the force of the B field on the wire and cause it to shrink at the given rate. Your first, assumption, however I believe is correct. That was how I originally read it, don't know why, but I skimmed over it.

 

[jtbell]

So I know what force each single piece will experience, which seems close to the answer but I seem to not be able to make that step. Integrating this over the loop will give 0 at each moment which is logical since the center of mass isn't moving.

Right, the net force on entire loop is zero. It seems to me the only way to give a sensible answer to "What force is needed...?" is as force (magnitude) per unit length along the wire. The question could have been phrased more precisely.

 

[haruspex]

It seems to me the only way to give a sensible answer to "What force is needed...?" is as force (magnitude) per unit length along the wire. The question could have been phrased more precisely.

I feel it is far more likely it means tension in the wire. E.g. the loop consists of a length of wire formed into a loop passing through a small ring. Pulling on the wire ends contracts the loop. Where the two strands pass through the loop they are in contact, hence the resistance of the loop is proportional to the circumference.

 

[rude man]

Eh? Wouldn't that depend on some elastic modulus of the wire? You can compute the force dF(t) = Bi(t)r(t)dθ pulling the loop inwards on all parts of the loop, 0 < θ < 2π, where i(t) = πB d/dt{r²(t)}/R(t) but then the deformation would depend on how stiff the loop wire is, would it not?

 

[haruspex]

Wouldn't that depend on some elastic modulus of the wire?

I was assuming it was inelastic.

 

[Coffee_]

Thanks for the replies all. I agree the question is worded not clearly but I just took it over from how I heard it. I thought that maybe to someone who already once solved a similar problem it would make more sense.

 

[haruspex]

Thanks for the replies all. I agree the question is worded not clearly but I just took it over from how I heard it. I thought that maybe to someone who already once solved a similar problem it would make more sense.

Fwiw, if my interpretation is correct (tension in the wire), I get that it's $-\frac{r \dot rB^2}{\rho}$

 

[Coffee_]

Fwiw, if my interpretation is correct (tension in the wire), I get that it's $-\frac{r \dot rB^2}{\rho}$

Would this be a legit way to find this tension? Take the circular loop in the x-y plane. Cut off the part where y<0. Then calculate the force along the z-axis from this half-circle. The parts that connect this half piece to the other piece are the ones that keeps this force at bay and thus that's the tension up to a factor 2?

 

[haruspex]

Would this be a legit way to find this tension? Take the circular loop in the x-y plane. Cut off the part where y<0. Then calculate the force along the z-axis from this half-circle. The parts that connect this half piece to the other piece are the ones that keeps this force at bay and thus that's the tension up to a factor 2?

Yes, that should work, but I think it's a little easier dealing with a small arc $d\theta$.