Problem 45 - Fundamentals of Physics

Question

A \(1250 \mathrm{~W}\) radiant heater is constructed to operate at \(115 \mathrm{~V}\). (a) What is the current in the heater when the unit is operating? (b) What is the resistance of the heating coil? (c) How much thermal energy is produced in \(1.0 \mathrm{~h}\) ?

Step-by-Step Solution

Answer

(a) 10.87 A; (b) 10.58 Ω; (c) 4,500,000 J

Step 1: Identify Given Information

You are given the power of the heater, \(P = 1250 \, \mathrm{W}\), and the voltage, \(V = 115 \, \mathrm{V}\). You need to find the current, \(I\), resistance, \(R\), and thermal energy produced in 1 hour.

Step 2: Calculate the Current

Use the power formula \(P = V \times I\) to solve for the current \(I\). Rearrange the formula to \(I = \frac{P}{V}\). Substitute the given values to find the current: \(I = \frac{1250}{115} \approx 10.87 \, \mathrm{A}\).

Step 3: Calculate the Resistance

Use Ohm's Law \(V = I \times R\) to find the resistance \(R\). Rearrange the formula to \(R = \frac{V}{I}\). Substitute the known values: \(R = \frac{115}{10.87} \approx 10.58 \, \mathrm{\Omega}\).

Step 4: Calculate Thermal Energy Produced in 1 Hour

The thermal energy can be calculated using the formula \(E = P \times t\), where \(t\) is time in seconds. Since 1 hour is 3600 seconds, \(E = 1250 \times 3600 = 4,500,000 \, \mathrm{J}\).

Key Concepts

Electric Power

Electric power is the rate at which electrical energy is transferred by an electric circuit. It is typically measured in watts (W). In our example, the radiant heater has a power of 1250 W. Power in an electrical circuit can be found using the formula:

• \( P = V \times I \)

This formula shows that electric power is the product of voltage (V) and current (I). Essentially, it tells us how much work is being done by the electric current in every second. It is crucial to understand this concept because it explains how much energy in the form of heat or motion the appliance will deliver. As such, electric power is central to many household electrical needs, illustrating how appliances consume and deliver electrical energy effectively.

Current Calculation

Current, measured in amperes (A), is the flow of electric charge through a conductor. To calculate the current in the radiant heater, we use the formula:

• \( I = \frac{P}{V} \)

Given that the power \( P \) is 1250 W and the voltage \( V \) is 115 V, substituting into the formula gives:

• \( I = \frac{1250}{115} \approx 10.87 \, \mathrm{A} \)

This value indicates how many amperes of current the heater uses when operating. Understanding how to calculate current is important because it helps to ensure that electrical devices receive the appropriate power for safe and efficient operation. Moreover, knowing the current allows us to determine what size circuit breaker or fuse is needed to protect the circuit.

Resistance Calculation

Resistance is a measure of the opposition to the flow of current in an electrical circuit. It is measured in ohms (Ω). Ohm's Law is essential in calculating resistance and is given by the equation:

• \( V = I \times R \)

Rearranging for resistance gives:

• \( R = \frac{V}{I} \)

With a current of 10.87 A and voltage of 115 V, the resistance of the heating coil is calculated as:

• \( R = \frac{115}{10.87} \approx 10.58 \, \Omega \)

Resistance affects how much electric current will pass through the heater, contributing to how efficiently it operates. Lower resistance in heating elements usually means less energy loss as heat, allowing the device to heat up quicker.

Thermal Energy

Thermal energy is the energy that comes from heat. This is the energy used by devices like our radiant heater to warm a space. To find the amount of thermal energy produced by the heater in an hour, use the formula:

• \( E = P \times t \)

Here, power \( P \) is 1250 W and time \( t \) is 3600 seconds (since one hour is 3600 seconds):

• \( E = 1250 \times 3600 = 4,500,000 \, \mathrm{J} \)

Therefore, the heater produces 4,500,000 Joules of thermal energy in that time frame. Understanding thermal energy is vital because it pertains to how energy conversion processes in electric devices lead to tangible outcomes like heating. It's the concept that lets us evaluate how much energy is needed to achieve a certain amount of work, such as warming a room.