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**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.

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.