Relation between induction voltage, condenser and resistance

[tsopatsopa]

Homework Statement

Check out the attached image.
The electromagnetic field B is constant.
The rod is moving through the rail and an inductive emf V is produced on the closed loop.
Due to that, the condenser C has a voltage difference Vc.
Question:
Express the relation between the total emf produced V and the condenser's voltage difference Vc in function of C and the resistance r (check out the attached image).

Homework Equations

The Attempt at a Solution

Well, no idea at all.
I tried this:
The total emf produced V = Vc + voltage of the resistance Vr
As the condenser and the resistor are connected by conductors, they must have the same voltage, so Vc = Vr
Then, I will get V= 2Vc.
BUT, the question asks me to express the relation between V and Vc in function of C and r. What I got is a simple V=2Vc.

Please help.

 

[anathema]

Hello there,

Thank you for bringing this question to my attention. It seems like you have the right idea, but the equation you have written is not quite complete. Let me explain further.

First, let's define the symbols we are working with:
V = total emf produced
Vc = voltage difference of the condenser
Vr = voltage difference of the resistor
r = resistance of the resistor
C = capacitance of the condenser

Now, as you correctly stated, because the condenser and resistor are connected by conductors, they must have the same voltage. Therefore, we can write the equation Vc = Vr.

Next, we know that the total emf produced is equal to the sum of the emf produced by the condenser and the emf produced by the resistor. In other words, V = Vc + Vr.

Now, we can substitute Vr with Vc in the equation above, since we know that they are equal. This gives us V = Vc + Vc, or simply V = 2Vc.

However, this is not the complete answer. We need to express V in terms of C and r. To do this, we need to use the equation for the voltage difference of a capacitor, which is Vc = Q/C, where Q is the charge on the capacitor and C is the capacitance.

We also know that the current flowing through the resistor is equal to the current flowing through the capacitor, since they are connected in series. Therefore, we can use Ohm's law to write Vr = I*r, where I is the current flowing through the circuit.

Now, we can substitute these equations into our original equation for V, which gives us:
V = Vc + Vr
V = Q/C + I*r

But we know that Q = CVc, so we can rewrite this as:
V = CVc/C + I*r
V = Vc + I*r

And finally, we can substitute Vc = Vr into this equation, which gives us:
V = Vr + I*r
V = Vc + I*r

And there you have it! The relation between the total emf produced V and the condenser's voltage difference Vc in terms of C and r is V = Vc + I*r.

I hope this helps. Let me know if you have any further questions or need any clarification.[