Problem 21 - College Physics

Question

In an \(R-L-C\) series circuit, \(R=150 \Omega, L=0.750 \mathrm{H},\) and \(C=0.0180 \mu \mathrm{F}\). (a) What is the resonant frequency of the circuit in rad/s? (b) Suppose you replace the inductor with one that has an inductance of \(L=0.25 \mathrm{H}\). What value of capacitance would be needed in order for the resonant frequency to remain unchanged?

Step-by-Step Solution

Answer

(a) Use \( \omega_0 = \frac{1}{\sqrt{LC}} \). (b) Solve for new \( C' \) using \( L' \) and the same \( \omega_0 \).

Step 1: Understanding Resonant Frequency Formula

The resonant frequency \( \omega_0 \) of an R-L-C series circuit is given by the formula: \[ \omega_0 = \frac{1}{\sqrt{LC}} \] where \( L \) is the inductance and \( C \) is the capacitance. In this step, we'll use this formula to calculate the resonant frequency using the provided values of \( L \) and \( C \).

Step 2: Calculating Initial Resonant Frequency

Given \( L = 0.750 \) H and \( C = 0.0180 \times 10^{-6} \) F, plug these values into the formula for resonant frequency: \[ \omega_0 = \frac{1}{\sqrt{0.750 \times 0.0180 \times 10^{-6}}} \] Calculate the resonant frequency in radians per second.

Step 3: Substitute New Inductor Value

The inductance is now replaced with \( L' = 0.25 \) H. We need the new capacitance \( C' \) so that \( \omega_0 \) remains the same. The formula remains: \[ \omega_0 = \frac{1}{\sqrt{L'C'}} \] Solve for \( C' \) using the constant \( \omega_0 \) calculated earlier.

Step 4: Solve for New Capacitance

Using the value of \( \omega_0 \) calculated in Step 2, set \[ \omega_0 = \frac{1}{\sqrt{0.25 \times C'}} \] and solve for \( C' \). \[ C' = \left( \frac{1}{\omega_0} \right)^2 \times \frac{1}{0.25} \] Calculate the new value of \( C' \) after simplification.

Key Concepts

Resonant Frequency

The resonant frequency in an RLC circuit is a crucial concept. It determines the frequency at which the circuit will naturally oscillate with the highest amplitude. For an RLC circuit, this is the frequency where the reactive effects of the inductor and capacitor cancel each other out, causing the impedance to be minimum and the circuit to behave purely resistive.

The formula for resonant frequency \( \omega_0 \) in radians per second is \( \omega_0 = \frac{1}{\sqrt{LC}} \). Here, \( L \) is the inductance and \( C \) is the capacitance of the circuit. This formula highlights the inverse relationship between the frequency and the square root of the product of inductance and capacitance. This means,

• If inductance decreases, the resonant frequency increases.
• If capacitance decreases, the resonant frequency increases.

This knowledge is vital for designing circuits to perform optimally at specific frequencies. Adjusting \( L \) and \( C \) allows engineers to tune the circuit for particular applications such as radio receivers and filters.

Inductance

Inductance, denoted as \( L \) and measured in henrys (H), is a property of a circuit or component (typically a coil) that determines how much it resists changes in electric current passing through it. It's essentially the measure of a component's ability to resist changes in current due to its magnetic field.

• Inductors are used to store energy in their magnetic field and in applications where energy needs to be maintained.
• The value of an inductor influences the circuit's resonant frequency, as seen in \( \omega_0 = \frac{1}{\sqrt{LC}} \).

In our exercise, replacing the original inductor with a new one with \( L = 0.25 \) H requires recalculating the capacitance \( C' \) to maintain the same resonant frequency. This involves using the same formula but solving for the new \( C' \). This illustrates how adjusting inductance alone can significantly affect the circuit's characteristics.

Capacitance

Capacitance, given by \( C \) and measured in farads (F), describes a component's ability to store an electric charge. Capacitors store energy in the electric field between their plates and can release energy quickly when required.

In a resonant RLC circuit, the value of capacitance is essential in determining the circuit's natural frequency. A capacitor contributes to the circuit's ability to oscillate at that frequency by balancing the inductance.

• Capacitance affects how quickly a circuit can respond to changes in voltage.
• Decreasing capacitance increases the circuit's resonant frequency, and increasing it lowers the frequency.

In the exercise, we find that when the inductor's inductance changes, we need to adjust the capacitance \( C' \) to maintain the same resonant frequency. This shows how the interdependence of these components sets the stage for tuning and designing circuits in radio frequencies and signal processing.