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

In summary, we have discussed the problem of finding the time taken for a circular disc with angular velocity w to slow to half its initial speed, ignoring friction and with a resistance R connected between the centre and rim. Using the given information, we were able to express the energy dissipated in the resistance as a function of the induced emf and resistance, and also found an expression for the rate at which rotational energy of the disk is changing. This allows us to set up a differential equation and solve for the time it takes for the disc to slow down to half its initial speed.
  • #1
fayled
177
0
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 :)
 
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  • #2
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?
 
  • #3
TSny said:
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.
 
  • #4
OK. Can you also find an expression for the rate at which rotational energy of the disk is changing?
 
  • #5
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.
 
  • #6
TSny said:
OK. Can you also find an expression for the rate at which rotational energy of the disk is changing?

dErot/dt=0.5ma2w(dw/dt)?
 
  • #7
Yes.
 

1. What is a homopolar generator and how does it work?

A homopolar generator is a type of electrical generator that produces direct current (DC) electricity. It works by using a magnetic field to induce a current in a conductor that is rotating within the field. This creates an electromagnetic force (EMF) that generates electricity.

2. What is the difference between homopolar and traditional generators?

The main difference between homopolar and traditional generators is the way they produce electricity. Homopolar generators use a stationary magnetic field and a rotating conductor, while traditional generators use a stationary conductor and a rotating magnetic field. Additionally, homopolar generators produce DC electricity, while traditional generators produce AC electricity.

3. How is the electromagnetic force (EMF) created in a homopolar generator?

The electromagnetic force (EMF) in a homopolar generator is created by the interaction between the magnetic field and the current in the rotating conductor. As the conductor rotates, the magnetic field lines cut through it, creating a change in magnetic flux which induces an electric current in the conductor. This current then produces the EMF, which generates electricity.

4. What are the main uses of homopolar generators?

Homopolar generators are mainly used in high-power applications, such as in particle accelerators and large-scale industrial motors. They are also used in some experimental and research projects, as well as in certain types of electric vehicles.

5. What are the advantages and disadvantages of using a homopolar generator?

The main advantage of using a homopolar generator is its simplicity, as it has fewer moving parts compared to traditional generators. This makes it more reliable and less prone to mechanical failures. However, homopolar generators are limited in their power output and can only produce DC electricity, which may not be suitable for all applications. They also require a large and complex magnetic field, making them more expensive to build and maintain.

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