Question

Two light bulbs for use at \(110 \mathrm{~V}\) are rated at \(60 \mathrm{~W}\) and \(100 \mathrm{~W}\), respectively. Which has the filament with lower resistance?

Short answer

Answer: The 100 W light bulb has a lower resistance.

Step-by-step solution

Calculate resistance for the 60 W light bulb

To find the resistance of the first light bulb (\(60 \mathrm{~W}\)), we will rearrange the formula \(P = \frac{V^2}{R}\) to solve for resistance, giving us \(R = \frac{V^2}{P}\). Now, we will plug in the given data for the \(60 \mathrm{~W}\) light bulb: \(R_1 = \frac{(110 \mathrm{~V})^2}{60 \mathrm{~W}}\)

Calculate resistance for the 100 W light bulb

Similar to step 1, we will use the same formula to find the resistance for the second light bulb (\(100 \mathrm{~W}\)): \(R_2 = \frac{(110 \mathrm{~V})^2}{100 \mathrm{~W}}\)

Compare the resistances

Now that we have calculated the resistance for both light bulbs, we can compare them to see which one has the lower resistance. We have: \(R_1 = \frac{(110 \mathrm{~V})^2}{60 \mathrm{~W}}\) \(R_2 = \frac{(110 \mathrm{~V})^2}{100 \mathrm{~W}}\) Since the resistance is inversely proportional to the power, the light bulb with higher power will have lower resistance. So, the light bulb with \(100 \mathrm{~W}\) power rating has the lower resistance.

Key concepts

Power Rating

When purchasing or using electrical devices, you'll frequently encounter their power rating. Power rating signifies how much energy a device consumes or how much work it performs within a certain period. It is typically measured in watts (W). The power rating of an electrical device allows us to understand its efficiency and energy consumption. For instance, a 60 W light bulb consumes 60 watts of power consistently when operating. Similarly, a 100 W bulb uses 100 watts. This rating is crucial because it tells us:

• How much power the device uses.
• Allows for the comparison of energy usage between devices.
• Helps manage electricity costs.

In applications where reducing energy consumption is vital, choosing devices with a lower power rating can lead to significant savings. Lower power ratings are generally more efficient for maintaining lower electricity bills.

Ohm's Law

Ohm's Law is a fundamental concept in electrical engineering that helps us understand the relationship between voltage, current, and resistance in a circuit. It states that the current through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. Mathematically, it's expressed as:\[ V = IR \]Where:

• \( V \) is the voltage.
• \( I \) is the current.
• \( R \) is the resistance.

Ohm's Law is useful because it enables us to solve for unknown values. If we know any two of the values (voltage, current, resistance), we can find the third. For instance, in the case of the light bulbs, although not directly using Ohm's law, the rearrangement formula \( R = \frac{V^2}{P} \) stems from the power formula integration into Ohm's principles, showing how intertwined these concepts are. Understanding Ohm's law helps you troubleshoot and design electrical circuits effectively.

Voltage

Voltage, often considered the 'electrical pressure,' is the force that pushes electric charges through a conductor. It's essential for the operation of any electrical device. Measured in volts (V), it specifies the potential difference in charge between two points in a circuit. Key aspects of voltage include:

• It's a measure of the energy per unit charge available to push electrons around a circuit.
• Necessary for creating a flow of charge or current.
• Higher voltages can push more electrons, resulting in higher currents.

In the light bulb example, both bulbs are rated at 110 V, meaning they're designed to work optimally at this voltage level. Knowing and maintaining the voltage rating is crucial because it ensures devices function properly and efficiently. Too low a voltage can result in insufficient power, while too high a voltage may damage devices due to the excessive flow of current.