Problem 81 - Fundamentals of Physics

Question

In a certain series \(R L C\) circuit being driven at a frequency of \(60.0 \mathrm{~Hz},\) the maximum voltage across the inductor is 2.00 times the maximum voltage across the resistor and 2.00 times the maximum voltage across the capacitor. (a) By what angle does the current lag the generator emf? (b) If the maximum generator emf is \(30.0 \mathrm{~V},\) what should be the resistance of the circuit to obtain a maximum current of \(300 \mathrm{~mA} ?\)

Step-by-Step Solution

Answer

a) Current lags by approx. 0° series dominance. b) Resistance should be 100 Ω.

Step 1: Understanding the relationships

Let's analyze the relationships given in the problem. We know that the maximum voltage across the inductor is twice that of the resistor and twice that of the capacitor. Let's denote: \(V_L = 2V_R = 2V_C\).

Step 2: Finding the phase relationship

The given problem implies a specific phase relationship in an RLC circuit driven at a certain frequency. In an RLC series circuit, at any given frequency, the current phase with respect to the generator's electromotive force (emf) \ can be determined from the ratio of its inductive and capacitive reactances. Here, the voltage across the inductor is equal to the voltage across the capacitor (given through the coefficients: twice both as the resistor), \ suggesting a resonance condition, so the current and voltage are in phase, implying the phase angle \(\phi = 0\). However, the current always lags the emf in case of the net inductive reactance, which seems to be minimized here, suggesting \(\phi = 0\) by net calculations. Hence, the current lags by \(90\) degrees due to inductor dominance, but appears as \(0\) degrees overall in this sense of comparison of dominance.

Step 3: Finding the resistance

Next, we determine the circuit resistance needed to achieve a maximum current of \(300 \text{mA}\) given a maximum generator emf of \(30.0 \text{V}\). At maximum current, the apparent resistance (impedance) accounts for the resistance in Ohm's Law: \(V = IR\), hence \(R_{total} = \frac{V_{max}}{I_{max}}\).Substituting the given values, \(R_{total} = \frac{30}{0.3} = 100 \Omega\).This gives us a standard circuit resistance value needed per total emf usage against the current capability.

Key Concepts

Inductive Reactance

In an RLC circuit, inductive reactance plays a crucial role in determining how currents and voltages react to an alternating current (AC) signal. Inductive reactance occurs because an inductor resists changes in current, creating a lag in the circuit. This lag manifests as a 90-degree phase difference between voltage and current.
When an AC signal passes through an inductor, it produces a back electromotive force (emf) due to the changing magnetic field around the inductor. This back emf opposes the current, leading to the concept of inductive reactance, calculated by the formula: \[ X_L = 2 \pi f L \] where \( X_L \) is the inductive reactance, \( f \) is the frequency, and \( L \) is the inductance of the coil.

• The higher the frequency or the inductance, the greater the inductive reactance.
• Inductive reactance is directly proportional to both frequency and inductance.

Understanding this helps us see why current would lag behind a generator's emf in an RLC circuit when the inductor's influence is prominent.

Capacitive Reactance

Capacitive reactance is a fundamental concept in understanding RLC circuit behavior. It describes how a capacitor impedes the flow of AC by storing and releasing energy in an electric field.A capacitor resists changes in voltage, leading to a 90-degree phase shift where the current leads the voltage.
The capacitive reactance can be calculated using the formula:\[ X_C = \frac{1}{2 \pi f C} \] where \( X_C \) represents the capacitive reactance, \( f \) is the frequency, and \( C \) is the capacitance.

• As the frequency increases, the capacitive reactance decreases, allowing more current to pass through.
• Unlike inductive reactance, capacitive reactance is inversely proportional to both frequency and capacitance.

In the context of our RLC circuit exercise, if the capacitive reactance equals the inductive reactance, an interesting state of resonance occurs, minimizing net reactance and affecting phase relationships.

Phase Angle

In AC circuits like the RLC, the phase angle \( \phi \) is key to understanding how current and voltage are related. This angle represents the extent to which the waveform of the current source (usually the generator's emf) is out of sync with the circuit's current waveform.
The phase angle is determined by the balance between inductive and capacitive reactances:\[ \phi = \arctan \left( \frac{X_L - X_C}{R} \right) \] where \( R \) is the resistance. A positive \( \phi \) means the circuit is more inductive and the current lags, while a negative \( \phi \) indicates a capacitive circuit where the current leads.

• A zero phase angle often indicates a resonant condition where inductive and capacitive reactances cancel each other out.
• This results in maximum current flow and minimal impedance.

For the given exercise, the condition mentioned implies the phase angle could be seen as zero in terms of net reactance, but reality shows slight lag due to inductive dominance—yet described as minimized here.