Problem 63 - College Physics

Question

A 500.0\(\Omega\) resistor is connected in series with a capacitor. What must be the capacitance of the capacitor to produce a time constant of 2.00 \(\mathrm{s} ?\)

Step-by-Step Solution

Answer

The capacitance should be 4.00 mF.

Step 1: Understand the Time Constant Formula

The time constant \( \tau \) for an RC (resistor-capacitor) circuit is given by the formula \( \tau = R \times C \), where \( R \) is the resistance and \( C \) is the capacitance. We are given \( \tau = 2.00 \) seconds and \( R = 500.0 \) ohms.

Step 2: Rearrange the Formula to Find Capacitance

We need to solve for \( C \) in the equation \( \tau = R \times C \). Rearranging the equation gives \( C = \frac{\tau}{R} \).

Step 3: Substitute Known Values

Substitute the known values into the equation: \( C = \frac{2.00}{500.0} \).

Step 4: Calculate the Capacitance

Perform the division to find the capacitance: \( C = \frac{2.00}{500.0} = 0.004 \) Farads or 4.00 milliseconds (4.00 mF).

Key Concepts

Time Constant

The time constant in an RC circuit is a crucial concept that describes how quickly the circuit responds to changes. It is labeled as \( \tau \) and defined as the time it takes for voltage across the capacitor to either charge up to about 63.2% of its maximum value or discharge to about 36.8% of its initial value. In mathematical terms, the time constant \( \tau \) is given by the product of resistance \( R \) and capacitance \( C \), which is expressed as \( \tau = R \times C \).
The time constant gives us an intuitive sense of how fast or slow the circuit is: a larger time constant means a slower response, while a smaller one means a faster response. This concept is significant for applications like timing and filtering in electronics, where precise control over charge and discharge rates is essential.

Capacitance Calculation

Calculating capacitance in an RC circuit involves using the relationship between time constant, resistance, and capacitance. If you are given the time constant \( \tau \) and resistance \( R \), you can find capacitance \( C \) using the formula \( C = \frac{\tau}{R} \).
In our example, with a time constant of 2.00 seconds and a resistance of 500.0 ohms, the capacitance can be calculated as follows:

• Start with the equation: \( C = \frac{\tau}{R} \)
• Substitute the values: \( C = \frac{2.00}{500.0} \)
• Perform the division: \( C = 0.004 \) Farads or 4.00 millifarads (mF)

This simple rearrangement and substitution process helps solve for unknown values and understand how these circuit components interact. Understanding this calculation helps in designing circuits with precise timing and charge retention.

Series Circuit

In a series circuit, components are connected end-to-end in a single path for the current to flow. This means each component shares the same current, but the voltage across the components adds up to the total voltage of the circuit. In our scenario, the resistor and capacitor are connected in series, which affects how they accumulate charge and discharge over time.
Series circuits are straightforward in terms of components' behavior:

• The total resistance in a series circuit is the sum of individual resistances: \( R_{total} = R_1 + R_2 + ... + R_n \).
• In terms of voltage, it is distributed across each component proportional to their resistances.
• Since all components share the same current, any alteration in one affects the entire circuit.

Understanding how series circuits function is critical for analyzing how the current and voltage interact, especially in RC circuits where timing and response are crucial.

Resistor

A resistor is a fundamental component in electronics that limits the flow of current in a circuit. In the context of an RC circuit, the resistor, alongside the capacitor, defines the time constant \( \tau \), influencing how quickly the circuit can charge or discharge.
Key roles of resistors in circuits include:

• Controlling current flow to prevent damage to other components.
• Creating voltage drops to supply other parts of the circuit with the proper voltage levels.
• Working with capacitors to shape signals in timing applications, such as clock generation or pulse shaping.

In the given exercise, a 500.0 \( \Omega \) resistor plays a critical role in determining the time constant, as demonstrated by its inclusion in the formula \( \tau = R \times C \). Understanding the function of resistors enhances your ability to manipulate electronic circuits for desired operational characteristics.