Problem 53 - Physics

Question

Two resistors, 42.0 and \(64.0 \Omega\), are connected in parallel. The current through the \(64.0-\Omega\) resistor is 3.00 A. (a) Determine the current in the other resistor. (b) What is the total power supplied to the two resistors?

Step-by-Step Solution

Answer

(a) Current in the 42.0 Ω resistor is approximately 4.57 A. (b) Total power supplied is approximately 1452.94 W.

Step 1: Determine the total voltage across the parallel circuit

In a parallel circuit, each component is subject to the same voltage. Using Ohm's law, the voltage across the 64.0 Ω resistor can be calculated as follows: \( V = I \times R = 3.00 \, \text{A} \times 64.0 \, \Omega = 192.0 \, \text{V} \). This is the voltage across both resistors.

Step 2: Calculate the current in the 42.0 Ω resistor

Using the known voltage (192.0 V) across the 42.0 Ω resistor, apply Ohm's Law to find the current: \( I = \frac{V}{R} = \frac{192.0 \, \text{V}}{42.0 \, \Omega} \approx 4.57 \, \text{A} \). This is the current through the 42.0 Ω resistor.

Step 3: Calculate the total current from the power supply

Since the resistors are in parallel, the total current is the sum of the currents through both resistors: \( I_\text{total} = 3.00 \text{ A} + 4.57 \text{ A} \approx 7.57 \text{ A} \).

Step 4: Calculate the total power supplied to the resistors

Total power supplied can be calculated using the formula \( P = V \times I_\text{total} \). Here, \( P = 192.0 \, \text{V} \times 7.57 \, \text{A} \approx 1452.94 \, \text{W} \). This is the total power supplied to the circuit.

Key Concepts

Parallel Circuits

Parallel circuits are a common way to connect electrical components, where each component has its own direct path to the voltage source. Unlike series circuits, where the current flows through each component sequentially, in parallel circuits each path operates independently.
In a parallel configuration:

• All components share the same voltage across them.
• The total current flowing into the circuit is the sum of the currents through each parallel branch.

This setup is particularly useful because it allows each component to be operated independently. If one component fails, the others continue to function as usual. In the context of the exercise, knowing that the voltage is identical along all paths simplifies the calculation by allowing us to use a single voltage value across different resistors.

Resistors

Resistors are components used in electrical circuits to control the flow of electric current. They resist or oppose the movement of electrical charges, which results in a reduction in current flow and a drop in voltage across the resistor.
The resistance of a resistor is measured in ohms (Ω), and it's a fundamental factor in Ohm's Law, which relates voltage (V), current (I), and resistance (R) in a circuit through the equation: \[V = I \times R\]In a parallel circuit, like the one described in the exercise, each resistor will have the same voltage across it, and individual currents can be calculated using the resistance value. For example, using the 64.0 Ω and 42.0 Ω resistors, Ohm’s Law allows us to know the current through each, once the voltage is established.

Electrical Current

Electrical current is the flow of electric charge, primarily transporting energy. It is measured in amperes (A), which indicates how much charge is flowing through a circuit per unit time.
In a parallel circuit, each branch has its own current, but all branches share the same voltage. The total current is the sum of the individual branch currents. Using Ohm’s Law, we calculate current through any resistor if either voltage or resistance is known:

• For the 64.0 Ω resistor, current is given as 3.00 A.
• For the 42.0 Ω resistor, we calculated the current as approximately 4.57 A.

Both these currents help us determine the total current flowing through the circuit when added together, providing insight into how the circuit operates overall.

Power Calculation

Power in an electrical circuit is a measure of how much energy is transferred by the circuit per unit of time, and it's measured in watts (W). The basic formula for calculating power is:\[P = V \times I\]Here, power depends on both voltage across the circuit and the current flowing through it. For parallel circuits, the power can also be computed by adding up the power used by each resistor separately, thanks to their shared voltage.
In the given exercise, once the total voltage (192.0 V) and total current (7.57 A) were computed, calculating the power became straightforward, yielding approximately 1452.94 W. This tells us the total energy consumption of all components in the circuit.