Problem 83 - Fundamentals of Physics

Question

An electric immersion heater normally takes 100 min to bring cold water in a well-insulated container to a certain temperature, after which a thermostat switches the heater off. One day the line voltage is reduced by \(6.00 \%\) because of a laboratory overload. How long does heating the water now take? Assume that the resistance of the heating element does not change.

Step-by-Step Solution

Answer

It will take approximately 113.2 minutes to heat the water.

Step 1: Understand the relationship between power, voltage, and time

The power of the heater is given by \( P = \frac{V^2}{R} \), where \( V \) is the voltage and \( R \) is the resistance. The time taken to heat the water is inversely proportional to the power, so if the power decreases, the time increases.

Step 2: Determine the effect of voltage reduction on power

When the voltage is reduced by \(6\%\), the new voltage \( V' = 0.94V \). The new power is \( P' = \frac{(0.94V)^2}{R} \). Simplifying gives \( P' = 0.8836 \frac{V^2}{R} = 0.8836P \). This means the power is reduced to \(88.36\%\) of its original value.

Step 3: Calculate the new time to heat the water

Initially, it takes 100 minutes to heat the water. Since power and time are inversely related, we use \( P \times t = P' \times t' \). Substitute the known values: \( 1 \times 100 = 0.8836 \times t' \). Solve for \( t' \): \[ t' = \frac{100}{0.8836} \approx 113.2 \] So, it now takes approximately 113.2 minutes to heat the water.

Key Concepts

Power Formula

The concept of electric power is important when discussing electrical appliances and circuits. Power is defined as the rate at which energy is transferred or converted per unit time. In electrical terms, the power used by an appliance is determined by the formula: \[ P = \frac{V^2}{R} \] where \( P \) is the power in watts, \( V \) is the voltage in volts, and \( R \) is the resistance in ohms. This formula reveals that power depends on the square of the voltage. If you raise the voltage, the power increases dramatically because the voltage is squared. Conversely, if the voltage decreases, the power drops significantly. This relationship is crucial to understanding how voltage changes affect power usage in practical scenarios like heating, lighting, and other appliances.

• Power measures how quickly energy is used
• The power formula shows a square relationship with voltage
• Changes in voltage can have a large impact on power

Voltage Reduction Effects

Voltage reduction has significant effects on electrical devices. When the line voltage provided to a device is reduced, the power output of that device also decreases. To see how this works, let's consider the previous situation with a voltage reduction of 6%. The new voltage after reduction is 94% of the original, or mathematically represented as \( V' = 0.94V \). Applying this change to our power formula: \[ P' = \frac{(0.94V)^2}{R} \] From this, we find that the power becomes \( 0.8836 \times P \), meaning the power output is reduced to 88.36% of the original. Consequently, because less power is being used, any task that the device is performing, such as heating, will take longer. Knowing how voltage fluctuations affect devices can help in planning and managing power effectively in both homes and industries.

• Voltage reduction decreases power usage
• Effect is compounded because voltage is squared
• Lower power results in longer operation times

Resistance in Circuits

In electrical circuits, resistance is the property that opposes the flow of electric current. Every circuit component, like wires or resistors, has some resistance. Resistance is measured in ohms \( \Omega \) and plays a vital role in determining how much current flows in a circuit for a given voltage. In our context, even if the resistance remains constant—as assumed in the original exercise—the total power output will still change if the voltage changes. This is because power depends on both voltage and resistance. It is worth noting that in real-world scenarios, factors such as temperature changes can affect resistance. However, for this example, assuming constant resistance simplifies the problem and focuses attention on voltage's influence.

• Resistance hinders electric current flow
• Measured in ohms and affects circuit functionality
• For this exercise, constant resistance highlights voltage's impact