Question

A voltage spike causes the line voltage in a home to jump rapidly from \(110 . \mathrm{V}\) to \(150 . \mathrm{V}\). What is the percentage increase in the power output of a 100.-W tungsten-filament incandescent light bulb during this spike, assuming that the bulb's resistance remains constant?

Short answer

Answer: Approximately 70.25%

Step-by-step solution

Find the initial current and resistance

We are given that the initial power P1 = 100 W and the initial voltage V1 = 110 V. Using the power formula: P1 = I1 * V1 We can find the initial current (I1) by rearranging the equation: I1 = P1 / V1 = (100 W) / (110 V) = 10/11 A Now, we can use Ohm's law to find the resistance (R) of the light bulb: V1 = I1 * R R = V1 / I1 = (110 V) / (10/11 A) = 121 Ω.

Calculate the new current and power during the voltage spike

Now that we have the resistance, we can use the new voltage (V2 = 150 V) to find the new current (I2) using Ohm's law: V2 = I2 * R I2 = V2 / R = (150 V) / (121 Ω) = 150/121 A Next, we will find the new power (P2) during the voltage spike using the Power formula: P2 = I2 * V2 = (150/121 A) * (150 V) = 22500/121 W

Calculate the percentage increase in the power output

The percentage increase in the power output during the voltage spike can be found using the following formula: Percentage increase = ((P2 - P1) / P1) * 100% Substitute the given values: Percentage increase = ((22500/121 W - 100 W) / 100 W) * 100% = ((8500/121 W) / 100 W) * 100% ≈ 70.25% Thus, during the voltage spike, there is an approximately 70.25% increase in the power output of the light bulb.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle used to understand how electric circuits behave. It defines the relationship between voltage (V), current (I), and resistance (R), stating that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance between them. The law is typically expressed with the formula:
\( I = \frac{V}{R} \)

When it comes to analyzing electric circuits, Ohm's Law is crucial. It allows us to calculate one of the three variables if we know the other two. For instance, in our exercise, we determined the resistance of the light bulb by using the initial voltage and current. This resistance remained constant throughout, which is an assumption based on the bulb’s properties not changing during the voltage spike.

Electric Power

Electric power quantifies the rate at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt (W), and power can be calculated using the formula:
\( P = I \times V \)

This formula is derived from the relationship between energy (E), power (P), current (I), and voltage (V), where power is the rate of energy consumption or generation over time. If we understand both the current flowing through a device and the voltage across it, we can calculate the electric power. In our exercise example, we initially know the power and voltage, so we could rearrange the formula to find the current. During the voltage increase, the exercise reveals how a spike in voltage can affect the power output, showcasing the direct proportionality between power and voltage.

Voltage Spike

A voltage spike is a sudden and sharp increase in voltage which can occur in electrical systems. These spikes are often transient or temporary and can be caused by various events such as lightning strikes, power outages, or turning on a large power-consuming device. Voltage spikes can be harmful to electrical equipment since they can exceed the designed operating voltage, potentially causing overheating or damage.

In our textbook exercise, the voltage spike from 110V to 150V plays a critical role in calculating the percentage increase in power output of a light bulb. A voltage spike of this nature raises the voltage across the light bulb, which according to Ohm's Law and the power formula, also increases the power — assuming the resistance does not change.

Resistance

Resistance is an electrical quantity that measures how the device or material reduces the electric current flow through it. Measured in ohms (Ω), resistance is crucial when analyzing circuits and their behavior. The resistance occurs due to collisions between electrons flowing through a conductor and the atoms of the conductor. These collisions convert some electrical energy into heat which is why resistive components like light bulbs heat up when in use.

In our exercise, the filament of the bulb provides resistance to the flow of electric current. Once the resistance is determined, as in step 1 of the solution, it remains constant for the purpose of the exercise. However, in real-life scenarios, resistance can vary with temperature, which can subsequently affect the current and power in a circuit. For simplicity, many problems assume a constant resistance to focus on the fundamental relationships between other quantities.