Solving Faraday's Disk: Induced Voltage & Max Current

[cdummie]

Homework Statement

Faraday's disk with radius of axle r1 and radius of disc r2 is shown in the picture below. Disc rotates in the homogenous magnetic field whose vector B is vertical to the disc. If disc is spinning 1000 rpm find induced voltage between r1 and r2 and what is maximal value of current through the resistor. Known values are: B=1T, r1=1cm, r2=20cm and R=10 Ω.

Homework Equations

e_ind=∫(v×B)dl

The Attempt at a Solution

E_ind=v×B=vBsin(v,B)

if there's 1000 rpm that is 16.7 rps and that is what i don't know, what to do next?

 

[rude man]

Homework Statement

Faraday's disk with radius of axle r1 and radius of disc r2 is shown in the picture below. Disc rotates in the homogenous magnetic field whose vector B is vertical to the disc. If disc is spinning 1000 rpm find induced voltage between r1 and r2 and what is maximal value of current through the resistor. Known values are: B=1T, r1=1cm, r2=20cm and R=10 Ω.

View attachment 87128

Homework Equations

e_ind=∫(v×B)dl

Just perform the integration from l = r1 to l = r2. What is v(l)?

 

[cdummie]

Just perform the integration from l = r1 to l = r2. What is v(l)?

Ok, this is what i have:

e_ind=∫(v×B)dl (i can't put limits of integration here but they are r1 lower and r2 upper limit)

E_ind=v×B=vBsin(v,B) , since there's no data about direction of the rotation i will assume it's counterclockwise (even though i think it doesn't matter because even if direction of ω is clockwise, angle between v and B would remain 90 degrees)
anyway,

E_ind=vB=ωrB since v=ωr, direction of E<sub>ind[/SUB is radial.

Now, back to eind

eind=∫ωrBdl*cos(Eind,dl) since both Eind and dl are radial (it also means i can write dl as dr) then cosine between them is 1, so i have:

eind=∫ωrBdr (again, limits of integration are: r1 lower and r2 upper)

eind=ωB(r²2 r²1)/2

Now, i have B and i have both r1 and r2 but i don't know how is the fact that disc is spinning 1000rpm is related to angular velocity (ω), that is the only thing missing, i mean, it's what i don't understand.

Now for the current through the resistor, it is just:

I=eind/R correct me if i made a mistake somewhere, but main problem is that i don't know the relation between speed of spinning and angular velocity.</sub>

 

[rude man]

E_ind=v×B=vBsin(v,B) , since there's no data about direction of the rotation i will assume it's counterclockwise (even though i think it doesn't matter because even if direction of ω is clockwise, angle between v and B would remain 90 degrees)

Right, but the polarity of the voltage will be determined by the direction of rotation.

Now, i have B and i have both r₁ and r₂ but i don't know how is the fact that disc is spinning 1000rpm is related to angular velocity (ω), that is the only thing missing, i mean, it's what i don't understand.

Angular velocity is radians per second. rpm is rotations per minute.
How many radians in 1 rotation?
How many seconds in 1 minute? You've done well so far, this should be a gimme. Like figuring out mph vs. ft per second!
You wrote v = ωr, you need to fully understand this relationship. Why is v = ωr?
Otherwise - good!
For one way to better understanding the physics, think of the disc as comprising a lot of thin radial wires each separated by thin insulation. The wires are thus all connected in parallel so the voltage is the same for all wires as they cut the flux lines of B. You learned emf = (B x l)⋅v I'm sure for a length of wire l. Least that's how I think of the Faraday disc.

 

[cdummie]

Right, but the polarity of the voltage will be determined by the direction of rotation.

Angular velocity is radians per second. rpm is rotations per minute.
How many radians in 1 rotation?
How many seconds in 1 minute? You've done well so far, this should be a gimme. Like figuring out mph vs. ft per second!
You wrote v = ωr, you need to fully understand this relationship. Why is v = ωr?
Otherwise - good!
For one way to better understanding the physics, think of the disc as comprising a lot of thin radial wires each separated by thin insulation. The wires are thus all connected in parallel so the voltage is the same for all wires as they cut the flux lines of B. You learned emf = (B x l)⋅v I'm sure for a length of wire l. Least that's how I think of the Faraday disc.

Well, in one rotation there's 2π radians, since one rotation means one full circle, and there's 60s in one minute, so i have to multiply value i have for rpm with radians and then divide it by 60. The way i think of relation v=ωr is that v is velocity as if the object that spins with angular velocity ω stops going around making circles because the whole system stops spinning and the object being there spinning along with the system (Faraday's disc in this case) just slips off, and the direction where it goes after that is the direction that vector v has at that point (this is the way i think of it). Since, in cases like this vectors r and ω are orthogonal it means that their cross product will give the direction of vector v at any point along the circle. Anyway, thanks a lot for the help!

 

[rude man]

Well, in one rotation there's 2π radians, since one rotation means one full circle, and there's 60s in one minute, so i have to multiply value i have for rpm with radians and then divide it by 60. The way i think of relation v=ωr is that v is velocity as if the object that spins with angular velocity ω stops going around making circles because the whole system stops spinning and the object being there spinning along with the system (Faraday's disc in this case) just slips off, and the direction where it goes after that is the direction that vector v has at that point (this is the way i think of it). Since, in cases like this vectors r and ω are orthogonal it means that their cross product will give the direction of vector v at any point along the circle. Anyway, thanks a lot for the help!

That sounds OK, at least your computation of ω is. But perhaps you're complicating the formula v = ωr. Think of a round race track, radius r. You're running around it with ω. Should be clear that your ground speed is v = ωr, ω in rad/s, r in meters, v in m/s.