How Does a Toroidal Inductor Affect Velocity in a Motional EMF Setup?

[dreadstar]

I have a variation of the Motional EMF in Tiplers Physics Second Edition 28-1. A conducting rod of mass m and length 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!

 

[lippyka]

The general equation for the EMF induced in a moving conductor is given by:EMF = -Blvwhere B is the magnetic field, l is the length of the conductor and v is the velocity of the conductor. In this case, the circuit includes a resistor connected to two conducting rails along which a rod of mass m and length l is sliding. The magnetic field is generated by a toroidal inductor of n turns per unit length and volume V= A*L. Assuming a velocity v(0) > 0 and small initial current I(0) > 0, we can use the above equation to calculate the EMF induced in the rod. The EMF induced will be given by:EMF = -Blv = -µnI(t)lv where µ is the permeability of the material, n is the number of turns per unit length, I(t) is the current at time t and l is the length of the rod. From this, we can calculate the force on the wire F = I(t)lB = µnI(t)l^2. This force will cause an acceleration a(t) = F(t)/m. Therefore, we can calculate the velocity of the rod v(t) as: v(t) = v(0) - µnI(t)l^2/m Since I(t) is changing with time, we can use the differential equation dv/dt = -(2µnl/m)I(t)dI/dt. This equation can be solved using the equation for the current through an LR circuit, dI/dt = -(R/L)I(t), where L is the inductance, which in this case equals µn^2V, and R is the resistance of the circuit. Therefore, we can solve the differential equation to get:dv/dt = -(2µnlR/m)I(t)^2. This equation can then be integrated to get v(t).

 

[M98Ranger]

Your approach to finding a function for v(t) is a good start. However, there are a few things to consider in your solution.

Firstly, the force on the wire should include the sine of the angle between the direction of the magnetic field and the direction of motion, as you mentioned in b). This would give you F = I*l*B*sin(theta).

Secondly, in your equation in b.1), you have both I(t) and I(0) which may lead to confusion. It would be clearer to use I(t) throughout the equation.

Thirdly, in your equation in f), you have used the inductance L as µ*n^2*V. However, in this case, the inductance is not a constant value but is dependent on the current flowing through the circuit. As the current changes, the magnetic field generated by the toroidal inductor also changes, affecting the inductance. So, you would need to use a variable for the inductance in your equation.

Regarding your question about accounting for the current source created by the combination of initial current and velocity, it is indeed possible to consider it as a capacitor or battery in the circuit. This would be similar to a situation where a battery is connected in series with a resistor and an inductor. In that case, the battery would provide a constant current which would then decrease as the inductor charges up, and the current would eventually reach a steady state. Similarly, in your case, the initial current would act as a source of current, and as the inductor charges up, the current would decrease until it reaches a steady state.

As for creating a circuit diagram, you could represent the initial current and velocity as a voltage source in series with a resistor, representing the combined effect of the initial current and velocity acting as a source of current. This would be connected in series with the toroidal inductor and the resistor in the circuit.

Overall, your approach is on the right track, but there are some adjustments that need to be made to account for the varying inductance and the angle between the magnetic field and the direction of motion. I hope this helps you in your further exploration of this problem.