Problem 14 - University Physics with Modern Physics

Question

A battery, a resistor, and a capacitor are connected in series in an RC circuit. What happens to the current through a resistor after a long time? Explain using Kirchhoff's rules.

Step-by-Step Solution

Answer

The behavior of the current through the resistor is that it becomes zero after a long time as the capacitor becomes fully charged. This is explained using Kirchhoff's Voltage Law (KVL) and Ohm's Law.

Step 1: Understanding the RC Circuit Components

An RC circuit consists of a resistor (R), a capacitor (C), and a battery. The resistor provides resistance to the current flow, the capacitor stores energy in an electric field, and the battery provides a constant voltage (V) across the circuit.

Step 2: Understanding Kirchhoff's Rules

Kirchhoff's rules consist of two rules that help us understand and analyze electric circuits. The first rule, Kirchhoff's Current Law (KCL), states that the total current entering a junction in a circuit must equal the total current leaving the junction. The second rule, Kirchhoff's Voltage Law (KVL), states that the sum of the voltage drops across all elements in a closed loop must be equal to the voltage supplied by the battery.

Step 3: Applying Kirchhoff's Voltage Law (KVL) to the RC Circuit

In an RC circuit, we can set up a loop consisting of the battery, the resistor, and the capacitor. According to KVL, the sum of the voltage drops across the resistor (V_R) and the capacitor (V_C) should be equal to the battery voltage (V). V = V_R + V_C

Step 4: Calculating the Voltage Drop across the Resistor (V_R)

The voltage drop across the resistor can be calculated using Ohm's Law, which states that the voltage across a resistor is directly proportional to the current flowing through it (I) multiplied by the resistance (R). V_R = I * R

Step 5: Calculating the Voltage Drop across the Capacitor (V_C)

The voltage across a capacitor can be expressed as: V_C = Q / C Where Q is the charge stored on the capacitor.

Step 6: Determining the Charge on the Capacitor after a Long Time

After a long time, the capacitor becomes fully charged, and the voltage across it (V_C) will be equal to the battery voltage (V). At this point, the capacitor acts as an open circuit, so no current will flow through it.

Step 7: Calculating the Current through the Resistor after a Long Time

Since the voltage across the capacitor (V_C) is equal to the battery voltage (V) after a long time, there will be no voltage drop across the resistor (V_R). Therefore, according to Ohm's Law (V_R = I * R), the current (I) through the resistor will be zero. In conclusion, after a long time, the current through the resistor becomes zero as the capacitor becomes fully charged. This behavior is explained using Kirchhoff's Voltage Law (KVL) and Ohm's Law.