Problem 59 - Physics

Question

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.

Step-by-Step Solution

Answer

The total current supplied by the source is approximately 9.22 A.

Step 1: Calculate the Series Resistance

When resistors are connected in series, the total resistance is the sum of their resistances. The coffee maker and frying pan are in series, so we sum their resistances: \[ R_{series} = 14 \, \Omega + 16 \, \Omega = 30 \, \Omega \]

Step 2: Calculate the Total Resistance

The series combination is in parallel with the bread maker's resistance. For resistors in parallel, use:\[ \frac{1}{R_{total}} = \frac{1}{R_{series}} + \frac{1}{R_{bread}} \] Plug in the values,\[ \frac{1}{R_{total}} = \frac{1}{30 \, \Omega} + \frac{1}{23 \, \Omega} \]This gives:\[ \frac{1}{R_{total}} = \frac{23 + 30}{690} = \frac{53}{690} \]Thus,\[ R_{total} = \frac{690}{53} \approx 13.02 \, \Omega \]

Step 3: Calculate the Total Current

Use Ohm's Law \( I = \frac{V}{R} \) to find the current. The voltage \( V \) across the source is 120 V. Using the total resistance from Step 2, substitute:\[ I = \frac{120 \, V}{13.02 \, \Omega} \approx 9.22 \, A \]

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits. It relates the voltage ( \( V \) ), current ( \( I \) ), and resistance ( \( R \) ) in a circuit through the equation:
\[ I = \frac{V}{R} \] This equation tells us that the current flowing through a circuit is directly proportional to the voltage and inversely proportional to the resistance. When you know any two of these quantities, you can use Ohm's Law to find the third.
Using Ohm's Law is straightforward:

• If you increase the voltage in the circuit and keep the resistance constant, the current will increase.
• If the resistance increases while the voltage remains constant, the current will decrease.

Understanding this principle is crucial as it forms the basis for more complex circuit calculations.

Series and Parallel Circuits

When dealing with circuits made of multiple components, it's important to recognize whether they are arranged in series or parallel.

**Series Circuits:**In a series circuit, components are linked one after the other, forming a single path for the current to flow. In our exercise, the coffee maker and frying pan are in series. The total resistance for series circuits is simply the sum of the individual resistances:
\[ R_{series} = R_1 + R_2 + ... + R_n \]This means that an increase in any of these resistances will increase the total resistance, affecting the current.

**Parallel Circuits:**Parallel circuits, on the other hand, offer multiple pathways for the current. Components are connected across the same voltage source. The bread maker in the exercise is parallel to the series combination of the coffee maker and frying pan.
The formula for calculating total resistance in parallel circuits is:\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \]As resistors are added in parallel, the total resistance decreases. Understanding how current and resistance work in these configurations helps in solving complex circuit problems.

Resistance Calculation

Calculating resistance is a key step in understanding how circuits behave. By being able to calculate total resistance, we can determine how much current will flow in a circuit given a certain voltage.

In the example provided, the coffee maker and frying pan create a combined resistance:

• The sum of their resistances ( \( 14 \Omega + 16 \Omega \) ) is \( 30 \Omega \) .
• This forms a series resistance.

Next, this series configuration is paralleled with another resistor (the bread maker with \( 23 \Omega \) ):
For parallel calculations:

• Use the reciprocal formula, \( \frac{1}{R_{total}} = \frac{1}{R_{series}} + \frac{1}{R_{bread}} \) , to find the total equivalent resistance.
• This gives us a total resistance of approximately \( 13.02 \Omega \) .

Accurate resistance calculation allows for precise predictions about circuit behavior and supports informed troubleshooting or design adjustments.