How Long Does It Take for a Homopolar Generator Disc to Halve Its Speed?

[fayled]

I have proven the induced emf between the centre and rim in a circular disc of radius a and angular velocity w, with a magnetic field B parallel to its axis is 0.5wa²B.

I need to find the time taken for the disc to slow to half it's initial speed ignoring friction, given a resistance R is connected between centre and rim and all other circuit resistance is negligible. It has mass m.

So I know it's moment of inertia is 0.5ma², and that energy is conserved such that the difference in the rotational KE initially and finally equals the energy dissipated in the resistance. This energy E is
E=0.5Iw²-0.5I(0.5w)²
E=0.5Iw²-0.125Iw²
E=(3/8)(0.5ma²)w²
E=(3/16)ma²w²

Now comes the problem in calculating the energy dissipated in the resistance. Obviously the induced emf and so induced current are time dependent. I'm not sure how I can get expressions for these in terms of time such that I could get the power dissipation with time and integrate for the energy. Any clues please? Thanks :)

 

[TSny]

Hello.

How would you express the rate at which energy is "consumed" in the resistor in terms of the emf ε and the resistance R?

 

[fayled]

Hello.

How would you express the rate at which energy is "consumed" in the resistor in terms of the emf ε and the resistance R?

dE/dt=ε²/R

Assuming this is correct, I can't see how to proceed to get the energy dissipated from there, because I don't know the dependence of ε with time.

 

[TSny]

OK. Can you also find an expression for the rate at which rotational energy of the disk is changing?

 

[BvU]

Hello Fayled,

difference in the rotational KE initially and finally equals the energy dissipated in the resistance

As you indicate, things are time dependent. So ω is a function of time. If you take small steps in time your difference becomes a differential and you end up with a simple differential equation.

If you are comfortable with that, go ahead. If not, write ΔKE = ... Δt with a function of ω on the dots. Also write ΔKE = ... Δω with a function of ω on the dots.

And: for our sake, also use the template after your first post.

 

[fayled]

OK. Can you also find an expression for the rate at which rotational energy of the disk is changing?

dE_rot/dt=0.5ma²w(dw/dt)?

 

[BvU]

Yes.