Problem 103 - Physics

Question

In the Arctic, electric socks are useful. A pair of socks uses a \(9.0-V\) battery pack for each sock. A current of \(0.11 \mathrm{~A}\) is drawn from each battery pack by wire woven into the socks. Find the resistance of the wire in one sock.

Step-by-Step Solution

Answer

The resistance of the wire in one sock is approximately 81.82 ohms.

Step 1: Understand the Problem

We are given a battery pack voltage and the current drawn by the wire woven into the socks. The goal is to find the resistance of the wire for one sock.

Step 2: Use Ohm's Law

Ohm's Law relates voltage (V), current (I), and resistance (R) in an electrical circuit. The formula is given as: \[ V = I \times R \]where \(V\) is the voltage, \(I\) is the current, and \(R\) is the resistance.

Step 3: Rearrange Ohm's Law for Resistance

To find resistance, rearrange the formula from Step 2 to solve for \(R\): \[ R = \frac{V}{I} \]

Step 4: Insert Given Values

Substitute the given voltage (\(9.0 \, \text{V}\)) and current (\(0.11 \, \text{A}\)) into the formula:\[ R = \frac{9.0}{0.11} \]

Step 5: Calculate the Resistance

Perform the division to find the resistance:\[ R = 81.82 \, \text{ohms} \]

Key Concepts

Resistance Calculation

Resistance is a measure of how much a material opposes the flow of electric current. In this context, we are interested in calculating the resistance of the wires within the socks. Using Ohm's Law, which states that voltage is the product of current and resistance, we can calculate resistance when voltage and current are known. The formula used for resistance calculation is:\[ R = \frac{V}{I} \]Here, the letter \( R \) stands for resistance, \( V \) for voltage, and \( I \) for current. Steps to Perform Resistance Calculation:

• Identify the known values: In our exercise, the voltage \( V \) is given as 9.0 V, and the current \( I \) is 0.11 A.
• Apply the formula: Substitute these values into the formula \( R = \frac{V}{I} \).
• Calculate: Perform the division to find the resistance; in this case, \( R = \frac{9.0}{0.11} \).

These steps show how resistance can be calculated from readily available measurements of voltage and current.

Electric Current

Electric current refers to the flow of electric charge through a conductor. It is a fundamental concept in understanding how electric circuits work. In the case of our socks, the current flows through the wires embedded in the fabric when the socks are connected to a battery pack. Key Characteristics of Electric Current:

• Measured in Amperes (A): Electric current is quantified using amperes, or "amps." For our socks, the current is given as 0.11 A.
• Direction of Flow: By convention, electric current flows from the positive to the negative terminal of a power source.
• Role in Circuits: The current is responsible for transferring energy from the power source (battery) through the circuit (the wire) to do work, like generating heat in the socks.

Understanding electric current is essential for comprehending how electrical devices function and how to safely manipulate circuits.

Voltage and Resistance Relationship

Voltage and resistance are two integral aspects of any electrical circuit. Understanding their relationship can help us predict how changes in one can affect the other, especially through Ohm's Law, which is pivotal in these calculations.The Relationship via Ohm's Law:Ohm's Law states:\[ V = I \times R \]This equation implies a direct relationship between voltage and current for a given resistance.

• Impact of Increased Voltage: If the resistance stays constant and the voltage increases, the current will increase by the same factor, assuming the material remains ohmic.
• Impact of Increased Resistance: Conversely, if resistance increases for a constant voltage, the current flowing through the circuit decreases, as seen in the equation \( I = \frac{V}{R} \).

By knowing how to use Ohm's Law, we can manage and design circuits more effectively by understanding how voltage, current, and resistance interact within them.