A series R-C combination is connected to an AC voltage of angular frequency omega = 500 rad/s. If th

Q: What is the time constant of an R-C circuit?

The time constant, denoted by $\tau$, is the product of the resistance and capacitance ($\tau = RC$). It represents the time required to charge the capacitor, through the resistor, to about 63.2% of its full charge.

Q: Why do we use a phasor diagram for AC circuits?

In AC circuits, voltages and currents are not always in phase. Phasor diagrams help us visualize these phase differences and calculate quantities like impedance using vector addition instead of complex trigonometric identities.

Q: How did we get $X_C = \frac{1}{\omega C}$?

This is the standard formula for the magnitude of capacitive reactance. It shows that the opposition a capacitor offers to AC current decreases as the frequency ($\omega$) or the capacitance ($C$) increases.

Q: Why is the impedance $Z = \sqrt{R^2 + X_C^2}$?

In a series R-C circuit, the voltage across the resistor is in phase with the current, while the voltage across the capacitor lags the current by 90 degrees. This orthogonal relationship means their combined opposition (impedance) must be calculated using the Pythagorean theorem.

Q: Can the time constant be negative?

No, both resistance and capacitance are positive physical quantities, so their product, the time constant, must always be positive.

Question

A series $R$-$C$ combination is connected to an AC voltage of angular frequency $\omega = 500 	ext{ rad/s}$. If the impedance of the $R$-$C$ circuit is $R\sqrt{1.25}$, the time constant (in millisecond) of the circuit is

Sigma Percentile · Accepted Answer

### Analyzing the Setup
Imagine you are looking at a simple electrical circuit with a resistor and a capacitor connected in series to an alternating current source. The problem gives us the angular frequency of the AC source, $\omega = 500 	ext{ rad/s}$, and a very specific relationship for the total impedance of the circuit: $Z = R\sqrt{1.25}$.

Our goal is to find the time constant of this circuit. But what exactly is the time constant? In an RC circuit, the time constant, denoted by $	au$, is simply the product of the resistance and the capacitance, $	au = RC$. So, our mission is to extract the value of $RC$ from the given impedance relationship.

### The Master Equation
To do this, we need to recall how impedance works in an AC circuit. Because the voltage across a resistor is in phase with the current, while the voltage across a capacitor lags by 90 degrees, we can't just add their oppositions algebraically. We have to add them as vectors!

This is where the phasor diagram comes in handy. The total impedance $Z$ is the hypotenuse of a right-angled triangle where the base is the resistance $R$ and the altitude is the capacitive reactance $X_C$. This gives us our master equation:
$$Z = \sqrt{R^2 + X_C^2}$$

### Solving for Reactance
Now, let's substitute the given value of $Z$ into our master equation:
$$R\sqrt{1.25} = \sqrt{R^2 + X_C^2}$$

To get rid of that pesky square root, let's square both sides:
$$1.25 R^2 = R^2 + X_C^2$$

Subtracting $R^2$ from both sides, we isolate the reactance term:
$$X_C^2 = 0.25 R^2$$

Taking the square root of both sides reveals a beautiful, simple relationship:
$$X_C = 0.5 R = \frac{R}{2}$$

### Final Calculation
We are almost there! We know that the capacitive reactance $X_C$ is defined as:
$$X_C = \frac{1}{\omega C}$$

Substituting this into our previous finding, we get:
$$\frac{1}{\omega C} = \frac{R}{2}$$

Now, let's rearrange this equation to isolate the time constant, $RC$:
$$RC = \frac{2}{\omega}$$

This is a fantastic result! The time constant depends only on the angular frequency, which we know is $500 	ext{ rad/s}$. Let's plug that in:
$$	au = \frac{2}{500} 	ext{ s}$$

$$	au = 0.004 	ext{ s}$$

To convert this to milliseconds, we multiply by 1000:
$$	au = 4 	ext{ ms}$$

And there we have it! By understanding the vector nature of impedance and the definition of the time constant, we've smoothly navigated to the final answer.

Animated Solution for Physics - Electromagnetic Induction: A series $R$ - $C$ combination is connected to an AC voltage of angular frequency $ω = 500 rad/s$ . If the impedance of the $R$ - $C$ circuit is $R 1.25$ , the time constant (in millisecond) of the circuit is

Enter Numerical Value:

Visualized Solution

$Z = R^{2} + X_{C}^{2}$

$Substitute Z$

$Square Both Sides$

$Solve for X_{C}$

$Express X_{C}$

$Time Constant τ$

$Final Calculation$

The Sigma Insight: Alternating Current (AC) and Voltage

Analyzing the Setup Imagine you are looking at a simple electrical circuit with a resistor and a capacitor connected in series to an alternating current source

The Master Equation To do this, we need to recall how impedance works in an AC circuit

Solving for Reactance

Final Calculation

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Enter Numerical Value:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Z=R2+XC2​​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substitute Z

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Square Both Sides

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solve for XC​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Express XC​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Time Constant τ

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Calculation