To tackle this problem, we need to analyze the behavior of an RC series circuit when subjected to alternating current (AC) at different frequencies. The key components to consider are the resistance (R) and the reactance (Xc) of the capacitor, which is frequency-dependent. Let's break this down step by step.
Understanding the Circuit Behavior
In an RC circuit, the total impedance (Z) can be expressed as:
Z = √(R² + Xc²)
Where Xc, the capacitive reactance, is given by:
Xc = 1 / (ωC)
Here, ω is the angular frequency, and C is the capacitance. The current (I) flowing through the circuit can be calculated using Ohm's law:
I = V / Z
Current at Frequency ω
At the original frequency ω, the current is I. Therefore, we can express this as:
I = V / √(R² + (1 / (ωC))²)
Current at Frequency ω/4
When the frequency is changed to ω/4, the new capacitive reactance becomes:
Xc' = 1 / ((ω/4)C) = 4 / (ωC)
Now, the total impedance at this new frequency is:
Z' = √(R² + (4 / (ωC))²)
The current at this frequency is given as I/2, so we have:
I/2 = V / √(R² + (4 / (ωC))²)
Setting Up the Equations
We now have two equations for the current:
1. I = V / √(R² + (1 / (ωC))²)
2. I/2 = V / √(R² + (4 / (ωC))²)
Relating the Two Equations
From the first equation, we can express V as:
V = I * √(R² + (1 / (ωC))²)
Substituting this expression for V into the second equation gives:
I/2 = (I * √(R² + (1 / (ωC))²)) / √(R² + (4 / (ωC))²)
By simplifying this, we can eliminate I (assuming I ≠ 0):
1/2 = √(R² + (1 / (ωC))²) / √(R² + (4 / (ωC))²)
Squaring Both Sides
Squaring both sides leads to:
1/4 = (R² + (1 / (ωC))²) / (R² + (4 / (ωC))²)
Cross-multiplying gives:
1/4 * (R² + (4 / (ωC))²) = R² + (1 / (ωC))²
Expanding and Rearranging
Expanding this equation results in:
1/4 R² + 4/(4ω²C²) = R² + 1/(ω²C²)
Rearranging terms leads to:
1/4 R² - R² + 4/(4ω²C²) - 1/(ω²C²) = 0
This simplifies to:
-3/4 R² + 4/(4ω²C²) - 1/(ω²C²) = 0
Finding the Ratio of Reactance to Resistance
Now, we can isolate R² and express the ratio of reactance to resistance:
3R² = 3/(ω²C²)
Thus, we find:
R² = 1/(ω²C²)
Now, substituting back to find the ratio of reactance (Xc) to resistance (R):
Xc = 1/(ωC)
Therefore, the ratio of reactance to resistance is:
Ratio = Xc/R = (1/(ωC)) / R = 1/(RωC)
This ratio gives insight into how the circuit behaves at different frequencies, illustrating the interplay between resistance and reactance in an RC circuit. The final answer is that the ratio of reactance to resistance at the original frequency ω is:
1/(RωC)
