Problem 25 - Applied Physics

Question

A digital timer is used on a 115 -V line. (a) If the resistance of the timer is $$12, \overline{0} 00 \Omega$$, how much current does it draw? (b) How much power does the timer use? (c) What does it cost to operate the timer at \(\$$ 0.09 / \mathrm{kWh}\) for 30 days?

Step-by-Step Solution

Answer

Current: 9.58 mA, Power: 1.1 W, Cost for 30 days: $0.071.

Step 1: Understanding the Resistance and Voltage Relationship

Ohm’s Law provides a way to calculate the current flowing through a circuit using the formula \( I = \frac{V}{R} \), where \( I \) is the current, \( V \) is the voltage, and \( R \) is the resistance. In our case, the resistance \( R = 12,000 \, \Omega \) and the voltage \( V = 115 \, \text{V} \).

Step 2: Calculating the Current Drawn by the Timer

Using Ohm's law, substitute the given values into the formula:\[ I = \frac{115}{12000} \approx 0.00958 \, \text{A} \]Thus, the current drawn by the timer is approximately \( 0.00958 \, \text{A} \) or \( 9.58 \, \text{mA} \).

Step 3: Understanding Power with Voltage and Current

The power consumed by an electrical device can be calculated using the formula \( P = VI \), where \( P \) is the power in watts, \( V \) is the voltage, and \( I \) is the current. We already calculated \( I = 0.00958 \, \text{A} \) and know \( V = 115 \, \text{V} \).

Step 4: Calculating Power Usage by the Timer

Use the power formula:\[ P = 115 \times 0.00958 \approx 1.1027 \, \text{W} \]The power used by the timer is approximately \( 1.1 \, \text{W} \).

Step 5: Understanding Energy Consumption and Cost

Energy consumption over time is measured in kilowatt-hours (kWh). The energy consumed is power multiplied by time. We will first calculate the energy usage over 30 days and then the cost using the rate \( \$0.09 / \text{kWh} \).

Step 6: Calculating the Energy Consumption

Convert power from watts to kilowatts and multiply by the number of hours in 30 days:\[ 1.1 \, \text{W} = 0.0011 \, \text{kW} \]\[ \text{Energy over 30 days} = 0.0011 \times 24 \times 30 = 0.792 \, \text{kWh} \]

Step 7: Calculating the Operational Cost

The cost of energy is:\[ \text{Cost} = 0.792 \times 0.09 = 0.07128 \]This cost is approximately \( \$0.071 \) to operate the timer for 30 days.

Key Concepts

Electrical Resistance

Electrical resistance is a key concept in understanding how electrical circuits work. It is a measure of how much a material resists the flow of electric current. The unit of resistance is the ohm (\(\Omega\)). A higher resistance means it's harder for the electricity to flow through the material.

In simple terms, think of resistance like a thin pipe in a water system. The thinner the pipe (or higher the resistance), the harder it is for the water (or electric current) to flow through. In our original exercise, a digital timer has a resistance of \(12,000 \, \Omega\).

• More resistance = less current flow
• Less resistance = more current flow

Understanding resistance is crucial for safely and effectively using electrical devices. It determines how much current will flow through an appliance when a given voltage is applied.

Electric Current Calculation

Electric current is the flow of electric charge and is measured in amperes (A). To calculate the current flowing through a circuit, we often use Ohm’s Law, which connects voltage (V), current (I), and resistance (R). Hatched as \(I = \frac{V}{R}\), this formula shows how current is directly proportional to voltage and inversely proportional to resistance.

Using the situation from our original exercise, given a voltage of 115 V and a resistance of \(12,000 \, \Omega\), the electric current is calculated as follows:

• Use the formula: \(I = \frac{V}{R}\)
• Substitute the values: \(I = \frac{115}{12,000} \approx 0.00958\)

This results in a current of approximately \(0.00958 \, A\), or \(9.58 \, \text{mA}\). This small amount of current shows the device operates on low power, contributing to safety and efficiency.

Power Consumption

Power consumption refers to the rate at which an electrical device converts electrical energy into another form of energy (like heat or light). It's expressed in watts (W). The power consumed can be found using the formula \(P = VI\), where \(P\) is power, \(V\) is voltage, and \(I\) is current.

In our example, using the values for current and voltage, the power used by the digital timer can be calculated:

• Use the formula: \(P = 115 \times 0.00958\)
• The result is approximately \(1.1 \, \text{W}\)

This calculation shows that the timer uses only about 1.1 watts of power. This level of power consumption is typical for small digital devices, making them energy efficient.

Energy Cost Calculation

Energy cost calculation helps you estimate how much it will cost to run an electrical device over a period. This involves calculating the total energy consumed and then applying the cost rate per kilowatt-hour (kWh).

To find out the energy cost for our digital timer:

• First, convert power in watts to kilowatts: \(1.1 \, W = 0.0011 \, \text{kW}\).
• Calculate the energy over 30 days: \(0.0011 \times 24 \times 30 = 0.792 \, \text{kWh}\).
• Finally, multiply by the energy cost rate: \(0.792 \times 0.09 = 0.07128\).

This results in an approximate cost of \(\$0.071\) for operating the timer for a 30-day period, offering a practical example of energy efficiency by keeping your energy costs low.