Problem 11 - College Physics

Question

In an \(R-L-C\) series circuit, the rms voltage across the resistor is \(30.0 \mathrm{~V},\) across the capacitor it is \(90.0 \mathrm{~V},\) and across the inductor it is \(50.0 \mathrm{~V}\). What is the rms voltage of the source?

Step-by-Step Solution

Answer

The RMS voltage of the source is 50.0 V.

Step 1: Understand the given circuit components

An R-L-C series circuit consists of a resistor (R), an inductor (L), and a capacitor (C). You are given the RMS (root mean square) voltages across these components: 30.0 V for the resistor, 90.0 V for the capacitor, and 50.0 V for the inductor.

Step 2: Apply Kirchhoff’s Voltage Law

In an R-L-C series circuit, the total voltage impressed across the circuit is not equal to the sum of the voltages across each component due to the phase differences. Instead, use Kirchhoff's Voltage Law which says the sum of the voltages in a closed circuit loop must be zero. However, for finding the source voltage in terms of the RMS values given, we use an equation considering these phase angles and magnitudes.

Step 3: Calculate the net reactive voltage

Calculate the net reactive voltage, which is the difference between the capacitive and inductive voltages (since these are 90 degrees out of phase with the resistor voltage and with each other). Subtract the inductor RMS voltage from the capacitor RMS voltage: \[ V_C - V_L = 90.0 ext{ V} - 50.0 ext{ V} = 40.0 ext{ V} \]

Step 4: Calculate the source voltage

To find the RMS voltage of the source, use the Pythagorean theorem for the vector sum of the resistor and net reactive voltages:\[ V_s = \sqrt{V_R^2 + (V_C - V_L)^2} \]Insert the values:\[ V_s = \sqrt{30.0^2 + 40.0^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50.0 ext{ V} \]

Step 5: Conclusion

We have determined that the RMS voltage of the source in the R-L-C series circuit is 50.0 V, calculated by taking the vector sum of the resistor's RMS voltage and the net reactive voltage.

Key Concepts

Kirchhoff's Voltage Law

Kirchhoff's Voltage Law is a fundamental principle used in circuit analysis. It states that the sum of the electrical voltages around any closed network is zero. In simpler terms, the total amount of energy gained and lost around a closed loop, such as in an RLC series circuit, must balance out. The reasoning behind this is that energy (or electric charge) cannot just appear or disappear in a loop; it is conserved and redistributed across the components. Each component in a circuit, such as resistors, inductors, and capacitors, has a voltage drop that contributes to this balance.

In the context of an R-L-C series circuit:

• The resistor causes a voltage drop in phase with the current.
• The capacitor causes a voltage drop that lags the current by 90 degrees.
• The inductor causes a voltage drop that leads the current by 90 degrees.

These differing phase angles mean that even though there might be large voltage drops across the capacitor and inductor, they don't add directly to the voltage across the resistor. This interplay of voltages is what Kirchhoff's Voltage Law helps us measure and calculate correctly.

Reactive Voltage

Reactive voltage refers to the voltages across reactive components, like inductors and capacitors, that do not consume power in the traditional sense. Instead, they store energy in the form of magnetic or electric fields. In an R-L-C circuit:

• The voltage across the capacitor is termed as capacitive reactance, and it is typically 90 degrees out of phase with the resistor voltage.
• The voltage across the inductor is termed as inductive reactance, also 90 degrees out of phase but in the opposite direction to the capacitor.

The interesting aspect is how these reactive voltages interact. Because they are 180 degrees apart (one leading and one lagging), we can find the net reactive voltage by subtracting one from the other. This net reactive voltage is considered at right angles to the resistive voltage, allowing us to use Pythagorean theorem to find the overall voltage in the circuit from these vector components.

In our example:

• The reactive voltage is computed as: \[ V_C - V_L = 90.0 \text{ V} - 50.0 \text{ V} = 40.0 \text{ V} \ \]
• This value indicates the net result of the interacting reactive components.

Root Mean Square Voltage

Root mean square (RMS) voltage is a method of expressing AC voltage. Unlike direct current (DC), alternating current (AC) voltage varies over time, making it challenging to define a constant voltage value. RMS voltage provides a way to express the effective value of this voltage, imagining it as if it were a DC voltage that delivers the same power to a resistor as the actual AC voltage does.

To find the RMS voltage in circuits, we often look at it as the square root of the means of the squares of the instantaneous values. This method provides a meaningful and practical way to determine the effectiveness of the voltage as it allows us to calculate power.

• The overall source voltage in the case of an RLC circuit can be determined using RMS values because they allow us to compute voltage contributions and power meanings effectively.
• In our example, the RMS source voltage was calculated by considering both resistive and reactive voltages: \[ V_s = \sqrt{V_R^2 + (V_C - V_L)^2} = \sqrt{30.0^2 + 40.0^2} = 50.0 \text{ V} \ \]

This formula represents how we can use RMS voltage to understand the combined effect of different types of voltage in a single expression and find the effective voltage of the circuit.