Problem 52 - Fundamentals of Physics

Question

An ac voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating emf of \(100 \mathrm{~V}\) (rms); the meter gives the same reading in volts in each case. What is this reading?

Step-by-Step Solution

Answer

The voltage reading is 100 V.

Step 1: Understanding the Problem

We have an AC series circuit comprising an inductor, capacitor, and resistor (LCR circuit) with the same voltage reading across each component. The circuit is connected to an alternating emf of 100 V (RMS). We need to find this common voltage reading.

Step 2: Concept of Impedance in LCR Circuit

In a series LCR circuit, the voltages across each component are equal due to the resonance condition, where the inductive reactance \(X_L\) equals the capacitive reactance \(X_C\). This results in the impedance of the circuit being purely resistive and equal to the resistance (R).

Step 3: Resonance Condition

At resonance, the impedance of the inductor (\(X_L = 2\pi f L\)) matches the impedance of the capacitor (\(X_C = \frac{1}{2\pi f C}\)), resulting in the cancellation of their reactive components. Only the resistive component of impedance remains, which equals the total applied voltage.

Step 4: Voltage Across Components

Since the voltmeter gives the same readings across the inductor, capacitor, and resistor, at resonance, the voltage reading across each of them is the same and equal to the supply voltage due to symmetry and identical impedance (100 V in this case).

Step 5: Final Calculation

Using the information from steps above, since resonance ensures equal impedance and therefore equal voltage division across each element, the voltage reading on the voltmeter across each component is 100 V.

Key Concepts

Resonance Condition

In an LCR circuit, the resonance condition is a special state where the circuit exhibits its natural frequency. This condition arises when the inductive reactance \(X_L = 2\pi f L\) matches the capacitive reactance \(X_C = \frac{1}{2\pi f C}\). When this occurs, the impedance due to the inductor equals the impedance due to the capacitor.
At resonance, the reactive effects of the inductor and capacitor cancel each other out.
This cancellation means that the circuit's impedance is purely resistive at this frequency.

• When a circuit is in resonance, it draws maximum power from the source.
• For students trying to remember this, think of resonance as the state where the circuit "hum" is at its natural sound frequency, making it efficient and functional.
• In the exercise, this condition meant that the voltage across each component was the same.

Impedance

Impedance is crucial in AC circuits and represents the total opposition a circuit presents to the flow of alternating current.
It is composed of both resistive and reactive elements.
In a series LCR circuit, impedance (\(Z\)) is defined as:

• \(Z = \sqrt{R^2 + (X_L - X_C)^2}\)
• Where \(R\) is the resistance, \(X_L\) is the inductive reactance, and \(X_C\) is the capacitive reactance.

In the context of our exercise, at resonance, \(X_L = X_C\) which simplifies the impedance to just the resistance \(Z = R\).
This simplification means that at resonance, the impedance is minimized to its lowest resistive value, allowing for maximal current flow.
Understanding impedance helps in predicting how alternating current will behave through the circuit.

Alternating Current

Alternating current (AC) is a type of electrical current where the flow of electric charge periodically reverses direction.
Unlike direct current (DC), which flows in a single direction, AC's changing direction makes it ideal for many applications.

• AC is used in power transmission due to its ability to travel over long distances.
• AC frequency, measured in hertz (Hz), indicates the number of times the current reverses direction per second.

This concept is significant in LCR circuits, influencing both the reactance and overall circuit behavior.
For the problem at hand, the alternating nature of the voltage source requires analyzing how each component of the circuit responds to these changes, especially under resonance conditions.
Being familiar with AC can greatly aid in predicting the responsiveness of circuits with reactive components.

Reactive Components

In an LCR circuit, reactive components are elements that store energy temporarily:

• Inductors (L) store energy in a magnetic field and have a reactance \(X_L\).
• Capacitors (C) store energy in an electric field and have a reactance \(X_C\).

These components do not dissipate energy like resistors but instead temporarily "hold" energy, which then affects the circuit's overall impedance.
In the exercise provided, the resonant condition aligns the reactance of these components, canceling them out.
This cancellation allows the power source to exclusively face resistive opposition.
Understanding reactive components is key to mastering the behavior of AC circuits, as they fundamentally shape the nature of impedance and current flow.