Transient current in inductance circuit

[krindik]

Homework Statement

I have posted my complete question http://imagebin.org/110458" [PLAIN]http://imagebin.org/110458. It is about a DC circuit with an inductor and a static magnetic field applied on a part of the circuit loop. An expression for the transient current needs to be found. My attempt is below. But I have doubt whether an EMF is created in the wire, where there is magnetic field.


2. The attempt at a solution


There is no energy conversion from the mechanical system to the electrical system. Also, the steady magnetic field does not effect the current but only creates a force. (an opposing current will be induced only if the magnetic field changes, AFAIK). So, we can consider the electrical and mechanical systems separately,

considering the electrical system,
the voltage drop is only across the inductance
$$V = - L \frac{di}{dt}$$
so, $$i = -Vt/L$$
However, I'm in doubt whether there is a voltage drop across the wire which is created as an opposing EMF due to the changing current, even though the magnetic field is fixed.
This is where I'm not sure about my attempt.



considering the mechanical system
$$\tau = (i \times B)R = iBR = I \frac{d\omega}{dt}$$
=> $$\omega$$ = integration of above

Also, we should be able to verify this by considering energy conservation
energy given by battery (Vi) - energy stored in the inductor (Li di/dt) - momentum gained by the wheel (Iw) + work done by the magnetic field (torque times integration of omega with time) = 0


Can somebody point me in the correct direction?

Thanks.

rgds,
K

 

[KataKoniK]

iran

Hello Kiran,

Thank you for posting your question. It looks like you are dealing with a very interesting circuit and I would be happy to help you find the correct direction for your solution.

Firstly, to address your doubt about whether an EMF is created in the wire where there is a magnetic field, the answer is yes. Even though the magnetic field is fixed, the changing current in the inductor creates a changing magnetic flux, which in turn induces an EMF in the wire. This EMF will oppose the change in current, which is why it is called an opposing EMF.

Now, let's take a closer look at your solution attempt. Your expression for the current is correct, but I would suggest using a negative sign in front of the Vt term to indicate the opposing EMF. So, your final expression for the current would be i = -Vt/L.

For the mechanical system, your calculation of the torque and angular velocity is correct. However, I would recommend using the equation for the energy stored in an inductor (E = 1/2 LI^2) to find the energy stored in the inductor instead of using Li di/dt. This will give you a more accurate representation of the energy conservation in the circuit.

Overall, your approach seems to be on the right track. Just remember to consider the opposing EMF in your calculations and use the correct equation for the energy stored in the inductor. I hope this helps and good luck with your solution!