Problem 1 - Applied Physics

Question

A soldering iron is rated at \(35 \overline{0} \mathrm{~W}\). If the current in the iron is \(4.00 \mathrm{~A}\), what is the resistance of the iron?

Step-by-Step Solution

Answer

The resistance of the soldering iron is approximately \( 2.19 \Omega \).

Step 1: Identify Given Values

We are given the power rating of the soldering iron, which is \( P = 35 \mathrm{~W} \), and the current flowing through it, \( I = 4.00 \mathrm{~A} \).

Step 2: Use Power Formula to Find Resistance

The power formula related to resistance and current is \( P = I^2 R \), where \( P \) is the power, \( I \) is the current, and \( R \) is the resistance. We will use this formula to find \( R \).

Step 3: Rearrange the Power Formula

Rearrange the formula \( P = I^2 R \) to solve for \( R \). This gives:\[ R = \frac{P}{I^2} \]

Step 4: Substitute Given Values into Formula

Substitute the known values into the equation: \( R = \frac{35 \mathrm{~W}}{(4.00 \mathrm{~A})^2} \).

Step 5: Calculate Resistance

Calculate the resistance:\[ R = \frac{35}{16} = 2.1875 \]Therefore, the resistance of the soldering iron is approximately \( 2.19 \mathrm{~\Omega} \).

Key Concepts

Electrical Resistance

Electrical resistance is a key concept in understanding how electric circuits operate. It refers to the opposition that a material offers to the flow of electric current. Essentially, resistance determines how much current will flow through a component when a voltage is applied across it. The unit of electrical resistance is the ohm, symbolized by \( \Omega \).

Resistance can depend on several factors:

• Material: Conductors like copper have low resistance, while insulators have high resistance.
• Length and cross-sectional area: Longer components have more resistance, and thicker components have less resistance.
• Temperature: As the temperature increases, the resistance of materials like metals typically increases.

In the example, we calculated resistance using the power and current values provided. Resistance tells us how much the soldering iron restricts electric current flow, which directly influences how much heat it generates.

Power in Electrical Circuits

In electrical circuits, power is the rate at which energy is used or converted in a circuit. It's often measured in watts (\( \mathrm{W} \)), which is a product of voltage (\( V \)) and current (\( I \)). The formula to calculate power is given by \( P = VI \).

For components where resistance is a factor, such as resistors or devices like a soldering iron, we also use the formula \( P = I^2 R \). This expresses power in terms of current and resistance, allowing calculations without needing the voltage.

Understanding power in circuits helps to determine:

• How much energy a device uses over time.
• How different values of current and resistance affect the energy consumed by a device.
• The efficiency and heat produced by electrical components.

In our example, the soldering iron is rated at 35 watts, indicating the power it consumes, which is key to calculating its resistance.

Electric Current Calculation

Electric current is the flow of electric charge through a conductor. Measured in amperes (\( \mathrm{A} \)), it represents how much charge passes through a point in the circuit per second. Current is central to understanding how electrical devices operate.

When calculating current, Ohm's Law is often used. It is expressed as \( I = \frac{V}{R} \), where \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance. This relationship shows that, for a constant voltage, increasing resistance will decrease the current.

In addition to Ohm's Law, calculating current can also involve the power formula when power and resistance are known: \( I = \sqrt{\frac{P}{R}} \).

In the context of the soldering iron, knowing the current allowed us to apply the power formula \( P = I^2 R \) to solve for resistance, demonstrating how current ties into calculating other significant electrical properties.