Heating a ring in a magnetic field

[Ananya0107]

Homework Statement

A ring of radius R is kept in the xy plane and a constant uniform magnetic field exists of magnitude B in the -k direction (negative z direction ) . It is heated through a temperature T . If the resistance of the ring is R₁find the final radius of the ring. Coefficient of linear expansion : α , mass of ring is m.

Note: this is not a question I picked up from a textbook, but I am confused about the outcome. I ask this question out of curiousity.

Homework Equations

E = -dφ/dt
r = R(1+αΔT) whereΔT is change in temperature.
dF = idl×B where dl here would be the differential length element that is heated.

The Attempt at a Solution

I calculated the magnitude of induced emf , as function of r,
E= B 2πr dr/dt,
I put dr/dt = α dT/dt, where T is absolute temperature
As this question is a conceptual doubt I assumed the absolute temperature T to be a function of t (a linear polynomial, T = at²+ bt +c
So I got the current in the ring as I = 2πR² B α (2at+b)/R₁
I knew that when we pull a straight rod through a magnetic field in a direction perpendicular to the magnetic field a mechanical force acts against the force of our hand equal to ilB where i = Blv/R where r is the resistance of the rod , the mass of the rod given to be m , it attains a terminal velocity which can be calculated by putting, a = ilB/m = dv/dt
So I thought that a retarding force would contract the ring, and when I checked ilB for the ring was radially inwards. Which made me think about the terminal velocity or terminal radius of the ring.
The confusion:
I could calculate the retarding acceleration of only the differential element dl , but how do I relate it to the whole ring . Also finally what equation of motion do i write for this element.
Please tell me if I am thinking in the right direction or if I should stop thinking about this problem if my approach is completely wrong.

 

[Simon Bridge]

You are thinking that you heat the ring, the ring expands, this let's more flux through the ring, which induces a current in the ring and the magnetic force opposes the expansion?

 

[Ananya0107]

Yes.

 

[Simon Bridge]

OK - the way to wrap your head around these sorts of things is to get rid of the complications you don't need.
i.e. why is it important that it is heating that causes the expansion - surely you just want to know about the interplay of forces when the loop expands?
Does the loop have t be a ring? Could it be a rectangle?
Does it need to expand uniformly, or could it have two stretchy sides so it can be elongated?