Question

Curve a in Fig. 31-21 gives the impedance Z of a driven RC circuit versus the driving angular frequency ${\mathit{\omega }}_{\mathbf{d}}$. The other two curves are similar but for different values of resistance R and capacitance C. Rank the three curves according to the corresponding value of R, greatest first.

Short answer

According to the value of resistance, the circuits are ranked as $Circuitc>Circuitb>Circuita$.

Step-by-step solution

The given data

1. The three circuits have different resistors and capacitors.
2. Curve b and curve c are similar.

Understanding the concept of impedance of a circuit

The impedance of the RC circuit is the effective resistance of an electric circuit or component to alternating current, arising from the combined effects of resistance and reactance. Thus, it depends on the resistance and the reactance of the capacitor. The reactance of the capacitor varies with the angular frequency of the oscillation.

Formula:

The impedance of a RC circuit,$\mathit{Z}\mathbf{=}\mathbf{}\sqrt{{\mathbf{R}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{1}}{{\left(\omega C\right)}^{\mathbf{2}}}}\mathbf{}$ (i)

Calculation of the ranking of the circuits according to their resistances

The analysis of the formula of equation (i) gives us the information that as $\omega$increases, the value of decreases $\frac{1}{{\left(\omega C\right)}^2}$. Thus, for higher values of ω, the capacitive reactance becomes less significant as compared to the resistance.

Thus, the graph of the impedance gives information about the resistance R at higher frequencies.

From the graph, we can see that at higher frequencies, the graph of circuit c has the highest resistance than circuit a and circuit b.

Hence, we rank the circuits in decreasing order of R as $Circuitc>Circuitb>Circuita$.