Question

A $200-\mathrm{\Omega }$ resistor, a $40.0-\mathrm{mH}$ inductor and a $3.0-\mu \mathrm{F}$ capacitor are connected in series with a time-varying source of emf that provides $10.0\mathrm{V}$ at a frequency of $1000\mathrm{Hz}$. What is the impedance of the circuit? a) $200\mathrm{\Omega }$ b) $228\mathrm{\Omega }$ c) $342\mathrm{\Omega }$ d) $282\mathrm{\Omega }$

Short answer

a) 200 Ohms b) 250 Ohms c) 300 Ohms d) 400 Ohms Answer: a) 200 Ohms

Step-by-step solution

Inductive reactance (XL)

Calculate the inductive reactance using the following formula: $XL=2\pi fL$ Where $f$ is the frequency and $L$ is the inductor value. Substitute the given values: $f=1000Hz$ and $L=40.0mH$. $XL=2\pi \times 1000\times 0.04=80\mathrm{\Omega }$

Capacitive reactance (XC)

Calculate the capacitive reactance using the following formula: $XC=\frac{1}{2\pi fC}$ Where $f$ is the frequency and $C$ is the capacitor value. Substitute the given values: $f=1000Hz$ and $C=3.0\mu F$. $XC=\frac{1}{2\pi \times 1000\times 3\times {10}^{-6}}=53\mathrm{\Omega }$ #step2# Calculate the total impedance of the circuit

Impedance formula

Apply the formula for the total impedance in a series RLC circuit: $Z=\sqrt{R^2+(XL-XC{)}^2}$ Where $R$ is the resistor value, $XL$ is the inductive reactance, and $XC$ is the capacitive reactance.

Substitute values

Substitute the given values and the calculated reactances into the impedance formula: $Z=\sqrt{(200\mathrm{\Omega }{)}^2+(80\mathrm{\Omega }-53\mathrm{\Omega }{)}^2}$ $Z=\sqrt{(200\mathrm{\Omega }{)}^2+(27\mathrm{\Omega }{)}^2}$

Calculate the impedance

Calculate the impedance using the substituted values: $Z=\sqrt{(200\mathrm{\Omega }{)}^2+(27\mathrm{\Omega }{)}^2}=\sqrt{40,729{\mathrm{\Omega }}^2}=202\mathrm{\Omega }$ Comparing the calculated impedance to the given options, the closest is: Answer: a) $200\mathrm{\Omega }$

Key concepts

Inductive Reactance

Inductive reactance, often symbolized as $X_L$, is a type of opposition that inductors produce to the flow of alternating current (AC) due to their ability to store energy in a magnetic field. It's not the same as resistance, which hinders both AC and direct current (DC). The value of inductive reactance depends on the frequency of the AC signal and the inductance of the component.

To calculate the inductive reactance, we use the formula $X_L=2\pi fL$, where $f$ is the frequency of the AC source in hertz (Hz), and $L$ is the inductance in henrys (H). Higher frequencies or larger inductors cause more reactance. In the given example, the inductive reactance at a frequency of 1000 Hz for a 40.0-mH inductor is calculated as follows:

$$X_L=2\pi \times 1000\times 0.04=80\mathrm{\Omega }$$
As frequency increases, the inductive reactance also increases, leading to a lower current through the circuit, as if the inductor were more 'resistant' to the AC flow.

Capacitive Reactance

Capacitive reactance, denoted $X_C$, is the measure of a capacitor's opposition to the change of voltage (and consequently, current) in an AC circuit. Unlike inductive reactance, it decreases with an increase in frequency or the capacitance value. The formula to calculate capacitive reactance is:$X_C=\frac{1}{2\pi fC}$, where $f$ is the frequency and $C$ is the capacitance.

For our problem, the capacitive reactance for a 3.0-\mu$F$ capacitor at a 1000 Hz frequency is:
$$X_C=\frac{1}{2\pi \times 1000\times 3\times {10}^{-6}}=53\mathrm{\Omega }$$
The capacitive reactance presents a unique property: it leads the current compared to the voltage across the capacitor. For the students to easily grasp this concept, it's essential to remember that in a purely capacitive circuit, voltage lags the current by ${90}^{\circ }$, the complete opposite of a purely inductive circuit, where voltage leads the current.

Series RLC Circuit

A series RLC circuit consists of a resistor (R), inductor (L), and capacitor (C) connected in a single path for the current to flow. In this type of circuit, you'll find a complex impedance arising from the combination of resistance and both kinds of reactance. The total impedance $Z$ in a series RLC circuit is calculated using the formula:$Z=\sqrt{R^2+(X_L-X_C{)}^2}$, which considers the combined effect of resistive and reactive (both inductive and capacitive) components.

The impedance is a complex quantity involving both magnitude and phase. However, in many cases, only the magnitude is of interest, which represents how much the circuit impedes the AC flow. In our example, the impedance calculation involves subtracting the capacitive reactance from the inductive reactance before squaring it and adding it to the square of the resistance value:
$$Z=\sqrt{(200\mathrm{\Omega }{)}^2+(80\mathrm{\Omega }-53\mathrm{\Omega }{)}^2}$$
Through our computation, we found the magnitude of the impedance to be approximately $202\mathrm{\Omega }$, which is closest to answer 'a', indicating that the series RLC circuit presents an opposition of about $200\mathrm{\Omega }$ to the AC source. It's important for students to understand that the total impedance in an RLC circuit will not simply be the sum of individual resistances and reactances due to the phase differences between them.