Problem 6 - Applied Physics

Question

A \(32.0-\Omega\) resistance coil uses \(375 \mathrm{~W}\) of power. What is the current in the coil?

Step-by-Step Solution

Answer

The current in the coil is approximately 3.424 A.

Step 1: Identify the Known Quantities

We know the resistance of the coil, \( R = 32.0 \, \Omega \), and the power used, \( P = 375 \, \mathrm{W} \). Our goal is to find the current in the coil, \( I \).

Step 2: Use the Power Formula for Current

The formula that relates power, current, and resistance is \( P = I^2 R \). To find the current \( I \), rearrange the formula to solve for \( I \).

Step 3: Rearrange the Power Formula

Rearrange the equation \( P = I^2 R \) to solve for \( I \) by dividing both sides by \( R \):\[I^2 = \frac{P}{R}\]

Step 4: Calculate the Square of the Current

Substitute the known values into the equation:\[I^2 = \frac{375 \, \mathrm{W}}{32.0 \, \Omega} = 11.71875\]

Step 5: Solve for the Current

To find \( I \), take the square root of \( 11.71875 \):\[I = \sqrt{11.71875} \approx 3.424 \mathrm{~A}.\]

Step 6: Confirm Unit Consistency

Ensure all units are properly accounted for and consistent. Resistance is in ohms (\( \Omega \)), power is in watts (\( \mathrm{W} \)), and current is in amperes (\( \mathrm{A} \)).

Key Concepts

Electric Current Calculation

Understanding electric current is crucial when analyzing circuits. The current, denoted by \( I \), is the rate at which electric charge flows through a conductor. You can think of it as how many electric charges pass a point in the circuit each second. It is typically measured in amperes, commonly known as amps (\( \, \mathrm{A} \)).
When calculating current, one essential formula is derived from Ohm's Law, which relates the voltage \( V \), resistance \( R \), and current \( I \) as follows: \( \displaystyle I = \frac{V}{R} \). However, when power (\( P \)) and resistance (\( R \)) are known, power in watts helps us find the current using the formula:

• \( P = I^2 R \)

To find \( I \), rearrange the formula to \( I = \sqrt{\frac{P}{R}} \). This equation means that you first divide the power by the resistance and then take the square root of the result to find the current. Calculating the current in electrical circuits ensures we understand how much charge is being transferred and allows us to analyze how well the circuit is functioning.

Power in Electrical Circuits

Power in electrical circuits is a measure of how much energy is being used or converted by the circuit per unit time. It's measured in watts (\( \, \mathrm{W} \)), where one watt is equal to one joule per second. In any electrical circuit, power can be understood using the formula \( P = IV \), which links power to current and voltage.
For resistive circuits like the one in our example, another useful expression for power involves the resistance \( R \) and current \( I \):

• \( P = I^2 R \)

This formula reveals the relationship between the power consumed by a resistor, the square of the current flowing through it, and its resistance. Knowing power is crucial because it helps us determine how much energy is being consumed and convert inside components like resistors. It aids in ensuring that components in a circuit are not subjected to excessive power that might damage them or that they can efficiently perform their functions without unnecessary loss of energy.

Resistance and Power Relationship

Resistance plays a vital role in determining how electrical components in a circuit use power. Resistance, measured in ohms (\( \, \Omega \)), is a property of a material or component that opposes the flow of electric current. Higher resistance means that less current can flow, given the same voltage.
The relationship between resistance and power can be observed in the formula:

• \( P = I^2 R \)
• \( P = \frac{V^2}{R} \)

This means that power dissipated as heat in the resistor will increase with a higher current flowing through it, provided the resistance remains constant. Conversely, as resistance increases while maintaining a constant voltage, the power decreases. This relationship highlights the importance of selecting the appropriate resistance in a circuit to ensure efficiency and safety. Components with too low resistance can allow too much current, causing overheating, while too high resistance can limit the circuit's effectiveness. Understanding this balance helps in designing circuits that are both safe and proficient in their energy use.