Homework Help: Variation on motional emf

1. Mar 13, 2004

I have a variation of the Motional EMF in Tiplers Physics Second Edition 28-1. A conducting rod of mass m and lenght l sliding along conducting rails connected by a resistor with a uniform magnetic field normal to the movement of the rod. The variation is that instead of just a resistor, the cicuit also includes a hypothetical toroidal inductor of n turns per unit length and volume V= A*L that is generating the magnetic field based on the current flowing through the circuit.

assume a velocity v(0) > 0 and small initial current I(0) > 0

I want to find a function for v(t).

i started with:
a) the B = µ*n*I for the field in the gap of a toroidial solenoid.
b) the force on the wire F=I*l*B (there may need to be sin theta in there)
b.1) because of a) F=µ*I(t)^2*l*n
c) i figured that a(t)= F(t)/m
d) thinking about a small delta t v(t) = v(0) - µ*I(t)^2*l*n/m
e) i then thought about dv/dt = -(2*µ*l*n/m) * I* dI/dt
f) from LR circuits we have dI/dt= -(R/L)*I where L is the inductance which equals µ*n^2*V where V is the volume of the toroidal solenoid.
g) replacing this into the dv/dt equation i got dv/dt = -(2*µ*l*n/m) * I* (R/µ*n^2*V)*I = -(2*µ*l*R*I^2)/(n*m)

In thinking about this as an LR circuit, I began to wonder about how to account for the current source created by the combinantion of the initial current and the velocity as a source of current and began to wonder if it is possible to picture the entire circuit with the velocity and initial current as a capacitor or battery. This would help me to confirm that i could use the defintion of dI/dt that i used in f).

A secondary goal would be to use the above idea to create a circuit diagram that would allow us to create a defintion for the current over time in such a configuration.

The initial current is really to initiate the magnetic field that would then generate a current defined by B*l*v/R

any help or feed back would be appreciated!

Last edited: Mar 13, 2004