An RLcircuit consists of a $40.0\Omega$resistor and a 3.00 mHinductor. (a) Find its impedance Z at 60.0 Hzand 10.0 kHz. (b) Compare these values of Zwith those found in Example 23.12 in which there was also a capacitor.

A circuit is a closed path via which electricity can flow from one location to another. It could include transistors, resistors, and capacitors, among other electrical components.

Resistance in the RL circuit is $R=40.0\Omega$

The inductor in the RL circuit is$L=3.00mH\left(\frac{{10}^{-3}H}{1mH}\right)=3.00\times {10}^{-3}H$

The formula for the determination of impedance of inductance can be expressed as,

$X_L=2\pi fL$………..(1)

Here $X_L$is inductive reactance, f is the frequency, and L is the inductance.

For each frequency, impedance can be expressed as,

$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$

………………(2)

At low frequency, f = 60 Hz, the value of $X_C=0\Omega$.

Substituting the given values in equation (1), we get

$$\begin{gathered} X_L=2\times 3.14\times \left(60Hz\right)\left(3.00\times {10}^{-3}H\right) \\ =1.13\Omega \end{gathered}$$

………………(3)

Then, using the given data and value from equation (3), the impedance can be calculated using equation (2), such that,

$$\begin{gathered} Z=\sqrt{\left({\left(40\Omega \right)}^2+{\left(1.13\Omega -0\Omega \right)}^2\right)} \\ =40.02\Omega \end{gathered}$$

At high-frequency $f=10.0kHz\left(\frac{{10}^{3}Hz}{1kHz}\right)=1.00\times {10}^{4}Hz$, substituting the given values in the above equation (1), we get

$$\begin{gathered} X_L^{\text{'}}=2\times 3.14\times \left(1\times {10}^{4}Hz\right)\left(3\times {10}^{-3}H\right) \\ =188.5\Omega \end{gathered}$$

Then, using the given data and value from equation (3), the impedance can be calculated using equation (2), such that,

$$\begin{gathered} Z^{\text{'}}=\sqrt{{\left({\left(40\Omega \right)}^2+\left(188.5\Omega \right)-0\Omega \right)}^2} \\ =192.7\Omega \end{gathered}$$

Therefore, the impedance at low and high frequencies are $188.5\Omega$and $193\Omega$.

Compare these values of Z with those found in the example 23.12 , in which there was also a capacitor.

As we can see, at f =60Hz , with a capacitor the value of impedance is, $Z=531\Omega$, that is about 13 times as high as without the capacitor.

The capacitor makes a large difference at low frequencies.

At f = 10 kHz , with a capacitor, the value of impedance is, $Z=190\Omega$, that is similar to the value obtained without the capacitor.

Thus, the capacitor has a smaller effect at high frequencies.