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Chapter 20: Problem 55

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

Short Answer

Expert verified
The current through the 42.0 惟 resistor is approximately 4.57 A, and the total power supplied is about 1452.94 W.

Step by step solution

01

Understanding the Parallel Circuit

In a parallel circuit, resistors are connected in parallel across the same two points. As a result, they have the same voltage across them. To solve the problem, we need to determine the voltage across each resistor and calculate the current in the other resistor.
02

Calculate Voltage across 64.0 惟 Resistor

Given the current across the 64.0 惟 resistor is 3.00 A, we can use Ohm's law to find the voltage: \[ V = I \times R = 3.00 \, \text{A} \times 64.0 \, \Omega = 192.0 \, \text{V}. \] This voltage is the same across the 42.0 惟 resistor because they are in parallel.
03

Determine the Current through the 42.0 惟 Resistor

Using Ohm's law, calculate the current through the 42.0 惟 resistor with the same voltage, 192.0 V: \[ I = \frac{V}{R} = \frac{192.0 \, \text{V}}{42.0 \, \Omega} \approx 4.57 \, \text{A}. \] This is the current through the 42.0 惟 resistor.
04

Calculate Total Current for the Circuit

The total current in the circuit is the sum of the currents through both resistors: \[ I_{\text{total}} = I_{42.0} + I_{64.0} \approx 4.57 \, \text{A} + 3.00 \, \text{A} = 7.57 \, \text{A}. \]
05

Calculate Total Power Supplied to Resistors

Use the formula for power, \[ P = V \times I_{\text{total}}, \] with the total current and the voltage across the resistors: \[ P = 192.0 \, \text{V} \times 7.57 \, \text{A} \approx 1452.94 \, \text{W}. \] Thus, the total power supplied is approximately 1452.94 watts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ohm's Law
Ohm's Law is a fundamental principle used to understand circuits, including parallel circuits. The law states that the voltage across a resistor is equal to the current flowing through it multiplied by its resistance. This relationship is expressed with the formula \( V = I \times R \), where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.
This simple formula helps us calculate unknowns such as voltage, current, or resistance if two of these quantities are known.
  • In the given exercise, Ohm's Law was used to determine the voltage across a 64.0 惟 resistor by multiplying the known current (3.00 A) with its resistance.
  • This calculated voltage, 192.0 V, was crucial since it is shared across both resistors in the parallel circuit configuration.
Resistor Calculations
In a parallel circuit, the same voltage is applied across all components. Resistor calculations are important when trying to find the current through different resistors in such configurations.
To find the current through a resistor in a parallel circuit:
1. Use the voltage across the resistors. Since the resistors are parallel, it will be the same for all. In this case, it was 192.0 V.
2. Apply Ohm's Law using this voltage and the resistance value of the resistor in question.
  • For the 42.0 惟 resistor, the current is calculated as \( I = \frac{192.0 \, \text{V}}{42.0 \, \Omega} \approx 4.57 \, \text{A} \).
  • By confirming these calculations, you ensure that the design of the circuit can accommodate the required current safely.
Electric Current
Electric current in a parallel circuit combines the currents through each path. Hence, to find the total current, you add the currents through the individual resistors.
  • In our exercise, we found the current through the 64.0 惟 resistor, known to be 3.00 A, and through the 42.0 惟 resistor, which we calculated to be approximately 4.57 A.
  • The total current flowing through the entire circuit is thus the sum of these two currents: \( I_{\text{total}} = 3.00 \, \text{A} + 4.57 \, \text{A} \approx 7.57 \, \text{A} \).
Understanding how to calculate total current is crucial. It helps in determining if a power supply can handle a circuit without being overloaded.
Power in Circuits
The power consumed or supplied to a circuit is important to measure as it indicates how much energy is being used. In circuits, power is calculated using the formula \( P = V \times I \), where \( V \) is the voltage and \( I \) is the total current in the circuit.
  • In our situation, the power supplied to the resistors was calculated using a voltage of 192.0 V and a total current of about 7.57 A.
  • This leads to power \( P \approx 192.0 \, \text{V} \times 7.57 \, \text{A} \approx 1452.94 \, \text{W} \).
The result, in watts, shows how much electrical energy is being converted to other forms of energy (such as heat) every second. Proper understanding of power calculations aids in designing efficient circuits and preventing potential hazards like overheating.

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Most popular questions from this chapter

A \(14-\Omega\) coffee maker and a \(16-\Omega\) frying pan are connected in series across a \(120-\mathrm{V}\) source of voltage. \(A 23-\Omega\) bread maker is also connected across the \(120-\mathrm{V}\) source and is in parallel with the series combination. Find the total current supplied by the source of voltage.

Nondigigital voltmeter A has an equivalent resistance of \(2.40 \times 10^{5} \Omega\) and a full-scale voltage of 50.0 \(\mathrm{V}\) . Nondigital voltmeter \(\mathrm{B},\) using the same galvanometer as voltmeter \(\mathrm{A}\) , has an equivalent resistance of \(1.44 \times 10^{5} \Omega\) What is its full-scale voltage?

A galvanometer has a full-scale current of 0.100 \(\mathrm{mA}\) and a coil resistance of 50.0\(\Omega\) . This instrument is used with a shunt resistor to form a nondigital ammeter that will register full scale for a current of 60.0 \(\mathrm{mA}\) . Determine the resistance of the shunt resistor.

A wire has a resistance of 21.0\(\Omega .\) It is melted down, and from the same volume of metal a new wire is made that is three times longer than the original wire. What is the resistance of the new wire?

An extension cord is used with an electric weed trimmer that has a resistance of 15.0\(\Omega .\) The extension cord is made of copper wire that has a cross- sectional area of \(1.3 \times 10^{-6} \mathrm{m}^{2} .\) The combined length of the two wires in the extension cord is 92 \(\mathrm{m} .\) (a) Determine the resistance of the extension cord. \((\mathbf{b})\) The extension cord is plugged into a \(120-\mathrm{V}\) socket. What voltage is applied to the trimmer itself?
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