Problem 8
Question
A heater operates on a \(11 \overline{0}\) - \(\mathrm{V}\) line and is rated at \(75 \overline{0} \mathrm{~W}\). What is the resistance of the heater element?
Step-by-Step Solution
Verified Answer
The resistance of the heater element is \(16.13\, \Omega\).
1Step 1: Understanding the Given Values
We know that the heater operates at a voltage of \(V = 110\, \text{V}\) and has a power rating of \(P = 750\, \text{W}\). These are the key values we will use to find the resistance.
2Step 2: Identifying the Formula to Use
Ohm's Law and the power formula relate power, voltage, and resistance. The formula we need is \(P = \frac{V^2}{R}\), where \(P\) is power, \(V\) is voltage, and \(R\) is resistance. We're solving for \(R\).
3Step 3: Rearranging the Formula
To find the resistance, rearrange the power formula: \(R = \frac{V^2}{P}\). This gives us the resistance in terms of known values of power and voltage.
4Step 4: Substitute Known Values into the Formula
Substitute \(V = 110\, \text{V}\) and \(P = 750\, \text{W}\) into the formula: \(R = \frac{110^2}{750}\).
5Step 5: Calculate the Resistance
Calculate \(R\) by performing the arithmetic: \(110^2 = 12100\), then \(R = \frac{12100}{750} = 16.13\, \Omega\).
Key Concepts
Electric ResistancePower FormulaVoltageElectric Heater Calculation
Electric Resistance
Electric resistance is a fundamental concept in electrical circuits. It represents the opposition to the flow of electric current through a conductor. Think of electric resistance as a barrier that makes it harder for electrons to flow through a wire. The greater the resistance, the lower the current for a given voltage. Resistance is measured in ohms (symbol: \( \Omega \)).
In practical terms, when you have a device like an electric heater, understanding its resistance can help explain how efficiently it operates. If the resistance is too high, the device might not heat effectively. Conversely, if the resistance is too low, it might draw too much current, potentially leading to overheating. Therefore, finding the right resistance is crucial for both efficiency and safety in electrical appliances.
In practical terms, when you have a device like an electric heater, understanding its resistance can help explain how efficiently it operates. If the resistance is too high, the device might not heat effectively. Conversely, if the resistance is too low, it might draw too much current, potentially leading to overheating. Therefore, finding the right resistance is crucial for both efficiency and safety in electrical appliances.
Power Formula
The power formula is a key equation that links the electrical power, voltage, and resistance in a circuit. Power (\(P\)) is the rate at which electric energy is transferred by an electric circuit. The unit for power is watts (symbol: \(W\)).
The power formula is expressed as \( P = \frac{V^2}{R} \), where \( V \) is voltage and \( R \) is resistance. This formula is a rearrangement of Ohm's Law and it beautifully shows how power increases with the square of the voltage or decreases as resistance increases if the voltage is constant.
Understanding the power formula helps determine the electrical power used by a device, which is crucial for calculating energy consumption, especially for high-power devices like heaters, which can influence electricity bills considerably.
The power formula is expressed as \( P = \frac{V^2}{R} \), where \( V \) is voltage and \( R \) is resistance. This formula is a rearrangement of Ohm's Law and it beautifully shows how power increases with the square of the voltage or decreases as resistance increases if the voltage is constant.
Understanding the power formula helps determine the electrical power used by a device, which is crucial for calculating energy consumption, especially for high-power devices like heaters, which can influence electricity bills considerably.
Voltage
Voltage is the electric potential difference between two points. It's like the pressure that pushes electric charges through a conductor. Measured in volts (symbol: \(V\)), it represents the energy per unit charge.
In our everyday lives, we encounter different voltage levels. For instance, household appliances typically operate at around 110V or 220V, depending on your country's standard.
In the context of an electric heater, knowing the operating voltage is critical. It tells you the potential energy available to make the heater work. If a heater requires a voltage of 110V, using it in a region with a different voltage supply may require an adapter or transformer to ensure safe and proper operation.
In our everyday lives, we encounter different voltage levels. For instance, household appliances typically operate at around 110V or 220V, depending on your country's standard.
In the context of an electric heater, knowing the operating voltage is critical. It tells you the potential energy available to make the heater work. If a heater requires a voltage of 110V, using it in a region with a different voltage supply may require an adapter or transformer to ensure safe and proper operation.
Electric Heater Calculation
Calculating the specifics of an electric heater, such as its resistance, involves understanding Ohm's Law and the power formula. Given a heater's voltage and power rating, you can determine its resistance using the formula \( R = \frac{V^2}{P} \).
For example, for a heater that operates at 110V with a power rating of 750W, substitute these values into the formula:\[ R = \frac{110^2}{750} \]Perform the calculation:\[ R = \frac{12100}{750} = 16.13 \, \Omega \]
This tells you that the resistance of the heater's element is 16.13 ohms, which is a valuable piece of information for ensuring that the heater operates safely and efficiently. Proper calculation of resistance helps optimize the operation, preventing issues like overheating or inadequate heating.
For example, for a heater that operates at 110V with a power rating of 750W, substitute these values into the formula:\[ R = \frac{110^2}{750} \]Perform the calculation:\[ R = \frac{12100}{750} = 16.13 \, \Omega \]
This tells you that the resistance of the heater's element is 16.13 ohms, which is a valuable piece of information for ensuring that the heater operates safely and efficiently. Proper calculation of resistance helps optimize the operation, preventing issues like overheating or inadequate heating.
Other exercises in this chapter
Problem 8
A circuit contains a \(575-\Omega\) resistance, a \(10 \overline{0}-\mu \mathrm{F}\) capacitor, and a \(0.400-\mathrm{H}\) inductance in series with a \(10 \ove
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Find the capacitive reactance of a \(6.00-\mathrm{mF}\) capacitor in a circuit of frequency \(10 \overline{0} \mathrm{~Hz}\).
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Find the apparent power produced by a generating station whose actual power is \(35 \overline{0}, 000 \mathrm{~kW}\) and whose power factor is \(0.860\).
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Find the resonant frequency of a circuit containing an \(8.00-\mu \mathrm{F}\) capacitor in series with a \(10 \overline{0}-\mu \mathrm{H}\) inductor.
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