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Homework Statement
A circular loop with resistance 0.5 [tex]\Omega[/tex] per meter of length is placed in a homogeneous magnetic field of induction B=2 Teslas. The magnetic field is perpendicular with the surface of the loop, as shown in the picture (image1). The radius of the loop is increasing according to the equation [tex] r(t)=vt[/tex] with [tex]v=0.5 m/s[/tex]. What is the induced voltage at time [tex]t=10 s[/tex]?
Homework Equations
U=Change in magnetic flux/change in time
The Attempt at a Solution
Well, it seems to me there are two ways of solving this problem: the Lorentz force method and Faraday's law method. I'll start with the Lorentz force method.
If we consider the fact that every part of the loop expands at the same rate, we can use the right hand rule for Lorentz force and find that the velocity of each part of the wire (and hence every electron) is perpendicular to the magnetic field, so:
[tex]Fm=Q*v*B[/tex]
At the same time, an electric field is generated by the stationary protons that is trying to pull the electrons back into place:
[tex]Fe=E*Q[/tex]
In the equilibrium position, the total force on the electrons will be zero:
[tex]EQ=Q*v*B[/tex]
And from this we get:
[tex]E=B*v[/tex]
Since:
[tex]E=U/d[/tex]
We obtain:
[tex]U=B*v*d[/tex]
Where d in this case is the circumference of the loop, so we plug that in and get:
[tex]U=2*r*B*v*\pi[/tex]
Since the radius changes with time, we plug that in as well:
[tex]U=2*v*t*B*v*\pi[/tex]
Since everything here is known, we put in all the values and get:
[tex]U=31.416 Volts[/tex]
Which is the correct answer according to the book (although I haven't really figured out for what purpose do I need the resistance per length value...).
The trouble I'm having is getting the other method to work properly.
I'm guessing the loop has an original nonzero size, so I'm going to use [tex]r_{0}[/tex] as that. At time [tex]t=0[/tex] the surface of the loop is:
[tex]S_{0}=r_{0}^{2}*\pi[/tex]
However, with time the radius gets larger:
[tex]r=r_{0}+vt[/tex]
Hence the surface gets larger as well:
[tex]S=r^{2}*\pi[/tex]
Since the surface of the loop changes, so does the magnetic flux:
[tex]\Delta\Phi=B*(SS_{0})[/tex]
Or:
[tex]\Delta\Phi=B*((r_{0}+vt)^{2}r_{0}^{2})*\pi[/tex]
Solving this as a difference of two squares:
[tex]\Delta\Phi=B*((r_{0}+vtr_{0})*(r_{0}+vt+r_{0}))*\pi[/tex]
[tex]\Delta\Phi=B*vt*(2*r_{0}+vt)*\pi[/tex]
So this is the part I don't follow: how am I supposed to use this method if the original radius is not specified? If I set it to be zero at time [tex]t=0[/tex] then I get half of the correct value. Am I missing something here?
P.S. Sorry if I made any latex errors, it took me quite a while to write all of this.
EDIT: Forgot to put the image in the attachment, think it works now.
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