Question

What is the time constant of a series RC circuit Mcq?

Answer 1

The time constant of a series RC circuit, which consists of a resistor (R) and a capacitor (C) connected in series, is a measure of how quickly the circuit responds to changes in voltage. It is defined as the time it takes for the voltage across the capacitor to either charge or discharge to approximately 63.2% of its final value.

The time constant \( \tau \) is given by the formula:

\[ \tau = R \cdot C \]

where:
- \( R \) is the resistance in ohms (Ω)
- \( C \) is the capacitance in farads (F)

In a multiple-choice question (MCQ) format, the options might look like:

A. \( R \cdot C \)

B. \( R + C \)

C. \( \frac{R}{C} \)

D. \( R - C \)

The correct answer is:

**A. \( R \cdot C \)**

Answer 2

In a series RC (resistor-capacitor) circuit, the time constant is a measure of how quickly the circuit responds to changes in voltage. It is defined as the product of the resistance \( R \) and the capacitance \( C \) in the circuit. The time constant, denoted by \( \tau \) (tau), determines how fast the voltage across the capacitor builds up or decays in response to a step change in voltage.

### Time Constant Formula

The formula for the time constant \( \tau \) in a series RC circuit is:

\[ \tau = R \cdot C \]

where:
- \( R \) is the resistance in ohms (Ω)
- \( C \) is the capacitance in farads (F)

### Behavior of the Time Constant

1. **Charging**: When the circuit is powered, the voltage across the capacitor \( V_C(t) \) as it charges follows the equation:

   \[ V_C(t) = V_{max} \left(1 - e^{-\frac{t}{\tau}}\right) \]

   where \( V_{max} \) is the final voltage of the power source, and \( t \) is the time elapsed.

2. **Discharging**: When the power source is removed and the capacitor discharges, the voltage \( V_C(t) \) across the capacitor decreases according to:

   \[ V_C(t) = V_{initial} \cdot e^{-\frac{t}{\tau}} \]

   where \( V_{initial} \) is the initial voltage across the capacitor at the moment discharge begins.

### Multiple Choice Question (MCQ) Example

Here's an example of an MCQ related to the time constant of a series RC circuit:

**Question:** In a series RC circuit with a resistor of 10 kΩ and a capacitor of 2 µF, what is the time constant of the circuit?

a) 20 ms  
b) 50 ms  
c) 10 ms  
d) 5 ms  

**Solution:**

To find the time constant \( \tau \), use the formula:

\[ \tau = R \cdot C \]

where:
- \( R = 10 \text{ kΩ} = 10,000 \text{ Ω} \)
- \( C = 2 \text{ µF} = 2 \times 10^{-6} \text{ F} \)

So:

\[ \tau = 10,000 \text{ Ω} \times 2 \times 10^{-6} \text{ F} \]
\[ \tau = 0.02 \text{ seconds} \]
\[ \tau = 20 \text{ ms} \]

**Answer:** a) 20 ms