Problem 11 - University Physics with Modern Physics

Question

If the capacitor in an \(\mathrm{RC}\) circuit is replaced with two identical capacitors connected in series, what happens to the time constant for the circuit?

Step-by-Step Solution

Answer

Answer: When a single capacitor in an RC circuit is replaced with two identical capacitors connected in series, the time constant for the circuit decreases by half.

Step 1: Understand the RC circuit and the time constant

An RC circuit is a simple electrical circuit that contains a resistor (R) and a capacitor (C) connected in series. The time constant (τ) of an RC circuit represents the time required for the voltage across the capacitor to either charge or discharge to about 63.2% of its final value. It is given by the equation: τ = RC.

Step 2: Determine the original time constant

Let's consider the original capacitor has capacitance C1, and the resistance of the circuit is R. The time constant for the original circuit, denoted as τ1, can be calculated using the formula τ = RC. Thus, τ1 = RC1.

Step 3: Find the equivalent capacitance for two capacitors in series

According to the problem, we replace the original capacitor with two identical capacitors connected in series. Let's denote their capacitances as C2 and C3. Since these capacitors are identical, C2 = C3 = C1. The formula for calculating the equivalent capacitance (Ceq) for capacitors connected in series is given by: 1/Ceq = 1/C2 + 1/C3 Plugging in the values of C2 and C3, we get: 1/Ceq = 1/C1 + 1/C1 = 2/C1 Now, we can find the equivalent capacitance: Ceq = C1/2

Step 4: Calculate the new time constant

Now that we have the equivalent capacitance for the two capacitors connected in series, we can calculate the new time constant (τ2) using the same formula as before, τ = RC. However, this time we will use the equivalent capacitance, Ceq: τ2 = R * (C1/2) = (1/2)RC1

Step 5: Compare the new time constant with the original time constant

To find the effect of replacing the single capacitor with two identical capacitors in series on the time constant, we need to compare the new time constant (τ2) with the original time constant (τ1): τ1 = RC1 τ2 = (1/2)RC1 From this comparison, we can conclude that the new time constant is half of the original time constant. So, when the capacitor in an RC circuit is replaced with two identical capacitors connected in series, the time constant for the circuit decreases by half.