Problem 39 - University Physics with Modern Physics

Question

Light Bulbs. The power rating of a light bulb (such as a \(100-\mathrm{W}\) bulb) is the power it dissipates when connected across a \(120-\mathrm{V}\) potential difference. What is the resistance of (a) a \(100-\mathrm{W}\) bulb and (b) a \(60-\mathrm{W}\) bulb? (c) How much current does each bulb draw in normal use?

Step-by-Step Solution

Answer

100-W bulb: 144 Ω, draws 0.833 A; 60-W bulb: 240 Ω, draws 0.5 A.

Step 1: Understanding the Formula for Power

The power rating of a light bulb, which is given in watts (W), is calculated using the formula: \[ P = V \times I \] where \( P \) is power, \( V \) is voltage, and \( I \) is current. Rearrange this formula to solve for current: \[ I = \frac{P}{V} \]

Step 2: Applying Ohm's Law to Find Resistance

Ohm's law states that \( V = I \times R \), where \( R \) is resistance. We can rearrange this to find resistance: \[ R = \frac{V}{I} \]. We'll first calculate the current for each bulb using the formula derived in Step 1, and then find the resistance.

Step 3: Calculating the Resistance of the 100-W Bulb

For the \(100\)-W bulb, we use the formula for current: \[ I = \frac{100\,\text{W}}{120\,\text{V}} = 0.833\,\text{A} \]. Then, use this current to calculate resistance: \[ R = \frac{120\,\text{V}}{0.833\,\text{A}} \approx 144\,\Omega \].

Step 4: Calculating the Resistance of the 60-W Bulb

For the \(60\)-W bulb, calculate the current: \[ I = \frac{60\,\text{W}}{120\,\text{V}} = 0.5\,\text{A} \]. Now, calculate the resistance: \[ R = \frac{120\,\text{V}}{0.5\,\text{A}} = 240\,\Omega \].

Step 5: Determining the Current For Each Bulb

The currents have already been calculated in Steps 3 and 4: - The \(100\)-W bulb draws \(0.833\,\text{A}\).- The \(60\)-W bulb draws \(0.5\,\text{A}\).

Key Concepts

Power Calculation

Power calculation in electrical circuits is an essential concept. Power, denoted by the symbol \(P\), is the rate at which electrical energy is transferred by an electric circuit. It is measured in watts (W).

• To calculate power, we use the formula \(P = V \times I\), where \(V\) stands for voltage in volts, and \(I\) stands for current in amperes (A).
• Here, voltage is the potential difference across the element, and the current is the flow of electric charges through it.

By rearranging the formula, we can also determine the current using the formula \(I = \frac{P}{V}\), which is particularly useful when we know the voltage and power, but need to find the current through the circuit. Breaking down this formula can greatly help in designing circuits and calculating the power requirements for various electrical devices.

Ohm's Law

Ohm's Law is one of the fundamental principles in the field of electronics. It establishes a relationship between voltage, current, and resistance in an electrical circuit. This law is described by the equation \(V = I \times R\).

• \(V\) is the voltage across the resistance.
• \(I\) is the current flowing through the resistance.
• \(R\) is the resistance, which opposes the flow of current.

By manipulating Ohm's Law, we can solve for any of the three variables if the other two are known. For instance, to find the resistance \(R\), we can rearrange the equation as \(R = \frac{V}{I}\). Understanding Ohm's Law allows us to handle and manage electrical circuits effectively, predicting how a circuit will behave under different conditions.

Electrical Resistance

Electrical resistance is a concept that describes how difficult it is for electric current to flow through a conductor. It is measured in ohms (Ω) and is determined by the material, length, and cross-sectional area of the conductor.

• The symbol for resistance is \(R\).
• In our light bulb example, resistance affects how much current flows for a given voltage.
• The higher the resistance, the less current flows at a given voltage.

Resistance can be calculated using Ohm's Law, \(R = \frac{V}{I}\). For example, for the 100-W bulb, given the voltage and current, we calculated the resistance to be approximately 144 Ω. Similarly, for the 60-W bulb, the resistance was calculated to be 240 Ω. Understanding resistance helps in designing circuits with optimal performance while preserving the integrity of electronic components.

Current Calculation

Current calculation is a vital part of analyzing and understanding electrical circuits. Current, denoted as \(I\), is the flow of electric charges through a conductor, and it is measured in amperes (A).

• The formula for calculating current is \(I = \frac{P}{V}\), where \(P\) is the power in watts, and \(V\) is the voltage in volts.
• In the given exercise, current was calculated for each bulb.
• For the 100-W bulb with a voltage of 120 V, the current is 0.833 A.
• For the 60-W bulb, it is 0.5 A, calculated using the same formula.

By calculating the current, we can evaluate how much electricity flows through a component of the circuit, which is crucial for ensuring that the components operate safely within their limits, avoiding overheating and ensuring energy efficiency. This concept, along with power and resistance, is key for effective circuit management.