Find an expression for the potential difference δvl across the inductor.

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Homework Equations
The Attempt at a Solution
Answers and Replies

There are 2 different potential differences in this question. One is the PD along each strip in the direction of $l$ - ie horizontal. I shall call this $V_x$. The other is the PD between the 2 strips in the direction of $b$ - ie vertical - at the same horizontal position $x$. I shall call this $V_y$, but note that like $V_x$ it also depends on $x$.

Option D is asking for an expression for $V_y$.

If the current has reached a limiting constant value, which will happen after the battery has been connected for a long time, then the PD $V_x$ along each strip depends only on its resistance not its inductance, because there is no back emf in an inductor if the current through it is not changing. In this case the vertical PD between the strips increases linearly from $0$ to $V_0$ when measured from the RH end where they are shorted : $$V_y=V_0 \frac{x}{l}$$

Before the current becomes constant there is a potential drop $V_R=IR$ across each strip due to its resistance $R$. Initially $I$ is very small so $V_R=IR \ll \frac12 V_0$. At the same time there is a back emf of $V_L=L\frac{dI}{dt}$ while the current increases, where $L$ is inductance. This is also a potential drop across the inductor, and it is linear (proportional to $x$) because inductance $L$ like resistance $R$ is proportional to length. Initially $\frac{dI}{dt}$ is large and $V_L =L\frac{dI}{dt} \approx \frac12 V_0$.

The sum of these 2 PDs equals the PD supplied by the battery, the same as when there is a separate resistor and inductor in series : $$V_R+V_L=\frac12 V_0$$ So the total PD $V_x$ along each strip varies linearly even when the current is changing, and the PD $V_y$ between the 2 strips again increases linearly from the RH end, as in the above equation.

In a circuit, a capacitor (Xc=30Ω, R=44Ω) and an inductor (Xl=90Ω, R=36Ω) are connected in series with an AC source (ƒ=60Hz, V=200V). Determine: 1-current intensity 2-the voltage across every component in the circuit.

Homework Equations

Z=√R²+(Xl-Xc) Where: Z is the impedance, Xl is the inductive reactance, and Xc is the capacitive reactance. I=V/Z

The Attempt at a Solution

That is how I solved the problem:

And that is how my text book solved it:

I get confused when I saw it, because I don't know why it consider the resistance of the capacitor to be an independant resistance while it didn't do with the resistance of the inductor, and determined the equivalent impedance of it, and consequently, it determined the potential difference across the inductor as 193.86V and not 180V as I did.

Could someone explain it for me?

Answers and Replies

cnh1995

This is how a lossy capacitor is represented:

But then Z is not 100 ohm as both the resistances are no longer in series.
Perhaps there is a mistake in the book's solution.

This is how a lossy capacitor is represented:
View attachment 112591But then Z is not 100 ohm as both the resistances are no longer in series.
Perhaps there is a mistake in the book's solution.

I've a feeling that they're not looking to introduce the complication of a parallel RC component. I'd judge from the solution method being used (dealing with reactances algebraically) that complex impedance hasn't yet been introduced in the course, and the circuit analysis would be fairly nasty and certainly not as simple as what the textbook indicates even allowing for an error on their part. But I do agree that the textbook solution is wrong in either case.

@Asmaa Mohammad is right to question the book's discrepancy in dealing with the resistance and reactance components differently for the two components. For the apparent course level I would be inclined to interpret the circuit as follows:

and to treat the capacitor and inductor in the same fashion mathematically.

Likes cnh1995

This is how a lossy capacitor is represented:
View attachment 112591But then Z is not 100 ohm as both the resistances are no longer in series.
Perhaps there is a mistake in the book's solution.

Actually, I didn't have an interpretation of a lossy capacitor in my assignment, I don't know why there are excerecises about it in my text book, may be for this the book treated its resistance as a separate resistance connected in series with the inductor and the capacitor.

@Asmaa Mohammad is right to question the book's discrepancy in dealing with the resistance and reactance components differently for the two components. For the apparent course level I would be inclined to interpret the circuit as follows:

and to treat the capacitor and inductor in the same fashion mathematically.

Gneill, still I don't know if my answer is correct or not?

cnh1995

Gneill, still I don't know if my answer is correct or not?

No. You should use your book's approach. Their calculation of inductor voltage is correct and they should have taken the 44 ohm resistor in series with the capacitive reactance. See gneill's diagram. That's what I meant when I said,

Perhaps there is a mistake in the book's solution.

Gneill, still I don't know if my answer is correct or not?

I would say it is not correct. I feel that the components of the circuit are the non-ideal capacitor and inductor and that their internal resistance can't be separated as separate components. The textbook's approach for the inductor is valid and they have obtained a correct result for it. My advice is to do the same thing for the capacitor.

When you submit your work for marking you might want to include a brief argument for not treating the resistance as separate components when judging the voltage across the "real" components.

Likes cnh1995

Their calculation of inductor voltage is correct and they should have taken the 44 ohm resistor in series with the capacitance

I would say it is not correct. I feel that the components of the circuit are the non-ideal capacitor and inductor and that their internal resistance can't be separated as separate components. The textbook's approach for the inductor is valid and they have obtained a correct result for it. My advice is to do the same thing for the capacitor.
When you submit your work for marking you might want to include a brief argument for not treating the resistance as separate components when judging the voltage across the "real" components.

Ok, guys, is this answer correct?

Likes gneill