Expert: James J. Kovalcin - 10/31/2009

Question
Imagine that you want to wind a solenoid that would be able to keep a 10-ohm light bulb lighted for longer than 1 s after the circuit is disconnected from the battery. (circuit is in series and current flows counterclockwise) If the form around which you wrap the coil has a radius of 5 cm, and you can wrap 2000 turns of superconducting (Zero resistance) wire around each linear meter of the form, how long will you have to make the form? Will this solenoid be approximately infinite (in the sense that its length is much greater than its radius)?
The other hint we received is that 1/3 of the initial current is equal to the minimum current to keep bulb lit. I can't even start it because I don't understand what its asking. The only thing I could think of was to use the equation for the magnetic flux.

Answer
The EMF generated by a changing magnetic flux is given according to Faraday's Law: EMF=-N*dF/dt where N is the number of turns in the inductor, dF is the change in the magnetic flux and dt is the time interval. The magnetic field through the core of a solenoid is given by B=muo*n*I where muo is the permeability of free space [muo=4*pi*10^-7] and n is the number of turns per unit length for the solenoid, and since magnetic flux F is equal to the product of the magnetic field B and the area A through which the field passes, Faraday's Law can be rewritten: EMF=-N*d[B*A]/dt=-N*d[muo*n*I*A]/dt.
Since the only thing changing here is the current this can be rewritten as:
EMF=-N*muo*n*A*dI/dt
Since the EMF will be equal to the product of the current and the resistance:
EMF=I*R=-N*muo*n*A*dI/dt
This then becomes a separable differential equation of the form:
-R/[muo*n*A*N]*dt=1/I*dI
Integrating both sides of the equation:
-R*t/[muo*n*A*N]+C=ln(I) where C is the constant of integration.
When t=0, I=Io and so the integration constant becomes:
C=ln(Io) where Io is the initial current.
The equation becomes:
-R*t/[muo*n*A*N]+ln(Io)=ln(I)
-R*T/[muo*n*A*N]=ln(I)-ln(Io)=ln[I/Io]
Rearranging the left side of the equation:
-(R/[muo*n*A*N])*t=ln[I/Io]
Raise both sides of the equation to the base e:
e^[-(R/[muo*n*A*N])*t]=I/Io
Solve for I:
I=Io*e^[-(R/[muo*n*A*N])*t]
If we take the time constant to be tc=muo*n*A*N/R:
I=Io*e^[-t/tc]
Since the problem says that 1/3 the initial current is the least current needed to light the bulb make the current I=Io/3:
Io/3=Io*e^[-t/tc]
Cancel out the initial current and take the natural log of both sides of the equation:
ln(1/3)=-1.099=-t/tc
Solve for the time constant where t=1sec [according to the problem statement]:
tc=t/1.099=1/1.099=0.91s
Now go back to our original definition of the time constant:
tc=0.91=muo*n*A*N/R
Now make a couple of substitutions.
"n" is the number of turns per unit length and is equal to:
n=N/L  therefore N=n*L
While A is the cross sectional area of the solenoid which is a circle:
A=pi*R^2
Substituting:tc=muo*n*A*N/R=muo*n*[pi*r^2]*[n*L]/R
Simplifying and solving for the length L:
L=R*tc/[muo*n^2*pi*r^2]=10*0.91/[4*pi*10^-7*2000^2*pi*0.05^2]=230m!
Whew!!!