AC voltage across a parallel branch

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

I am assuming that the given value of voltage applied is an RMS value .

Same voltage is applied across both the branches .

In the upper branch the net resistance is the sum of capacitive reactance of the capacitor and resistance of the resistor .

Using Impedance triangle , the net reactance is 100√2 Ohms and the net applied voltage across the branch is ahead of the voltage across the resistor by a phase π/4 . Hence voltage across the resistor is 20/√2 or 10√2 volts i.e option 3) .

This is also the given answer .

But I think option 2) is also correct . The net resistance of the upper branch is 100√2 Ohms and that of the lower branch is 50√2 Ohms .

The resistances in the two branches are in parallel .

The net resistance across the voltage source is (100√2)(50√2)/(100√2 + 50√2) = 10000/(150√2) Ohms

Current in the circuit = 0.3√2 Ohms . This makes option 2) also correct .

Is option 2) also correct along with option 3) ?

 

[gneill]

But I think option 2) is also correct . The net resistance of the upper branch is 100√2 Ohms and that of the lower branch is 50√2 Ohms .

The resistances in the two branches are in parallel .

The net resistance across the voltage source is (100√2)(50√2)/(100√2 + 50√2) = 10000/(150√2) Ohms

Current in the circuit = 0.3√2 Ohms . This makes option 2) also correct .

Is option 2) also correct along with option 3) ?

No, you can't combine resistor+reactance magnitudes as though they were resistances. It would be akin to "adding" vectors by their magnitudes alone.

Instead what you can do is use complex numbers to write the resistances and reactances in the form of impedances. Then you can combine these impedance values using the same formulas that you would for resistors.

 

[Jahnavi]

No, you can't combine resistor+reactance magnitudes as though they were resistances.

Are you implying that net resistance of the upper branch is 100√2 Ohms and that of lower branch is 50√2 Ohms but the two resistances are not in parallel ?

 

[gneill]

Are you implying that net resistance of the upper branch is 100√2 Ohms and that of lower branch is 50√2 Ohms but the two resistances are not in parallel ?

While you can say that the "net resistance" of the individual branches are what you've written, you cannot combine those values as if they were resistors. They are not pure resistances, but the magnitudes of the branch impedances.

The inductance and capacitance each present a reactance, not a resistance. Reactance not only "resists" current flow, but also produces a phase shift of the voltage developed across it versus the current flowing through it.

To conveniently work with reactive components we write the reactance as impedance using complex numbers. These impedance values can then be manipulated just as you would resistor values, the only difference being that you need to perform complex arithmetic. The complex math automatically takes care of dealing with the phase shifts that the components create.

 

[cnh1995]

Are you implying that net resistance **impedance of the upper branch is 100√2 Ohms and that of lower branch is 50√2 Ohms but the two resistances are not in parallel ?

The two resistances aren't in parallel, but the two impedances are.
As gneill mentioned, you need to represent the impedances in complex form.

 

[Jahnavi]

Thanks @gneill and @cnh1995 .