/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 93 A physics student uses a \(115.0... [FREE SOLUTION] | 91Ó°ÊÓ

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A physics student uses a \(115.00-\mathrm{V}\) immersion heater to heat 400.00 grams (almost two cups) of water for herbal tea. During the two minutes it takes the water to heat, the physics student becomes bored and decides to figure out the resistance of the heater. The student starts with the assumption that the water is initially at the temperature of the room \(T_{i}=25.00^{\circ} \mathrm{C}\) and reaches \(T_{f}=100.00^{\circ} \mathrm{C}\) The specific heat of the water is \(c=4180 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}} \cdot\) What is the resistance of the heater?

Short Answer

Expert verified
The resistance of the immersion heater is: \[R = \frac{\left(115.00\,\text{V}\right)^2}{\frac{\left(0.4\,\text{kg}\right)\left(4180\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\right)\left(75.00\,\text{K}\right)}{120\,\text{s}}} \approx 31.41\,\Omega.\]

Step by step solution

01

Calculate the Amount of Energy Required to Heat the Water

First, we need to determine the amount of energy required to heat the water by the given temperature. We can use the formula for heat transfer: \(Q = mc\Delta T\), where \(Q\) is the energy, \(m\) is the mass of the water, \(c\) is the specific heat, and \(\Delta T\) is the change in temperature. Given the mass of water \(m = 400.00\,\text{g} = 0.4\,\text{kg}\), the specific heat \(c = 4180\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\) and the initial and final temperatures \(T_i = 25.00^{\circ}\text{C}\) and \(T_f = 100.00^{\circ}\text{C}\), we can find the change in temperature \(\Delta T = T_f - T_i = 100.00 - 25.00 = 75.00\,\text{K}\) and then calculate the energy required: \[Q = mc\Delta T = \left(0.4\,\text{kg}\right)\left(4180\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\right)\left(75.00\,\text{K}\right).\]
02

Calculate the Power Used by the Heater

Now that we have the energy required to heat the water, we need to calculate the power used by the heater. We know that it takes 2 minutes to heat the water, which is equal to 120 seconds. We can find the power by dividing the energy required by the time taken: \[P = \frac{Q}{t},\] where \(P\) is the power, \(Q\) is the energy, and \(t\) is the time. So, we have: \[P = \frac{\left(0.4\,\text{kg}\right)\left(4180\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\right)\left(75.00\,\text{K}\right)}{120\,\text{s}}.\]
03

Calculate the Resistance of the Heater

Finally, we can use the formula for electrical power, \(P = \frac{V^2}{R}\), to find the resistance of the heater: \[R = \frac{V^2}{P},\] where \(R\) is the resistance, \(V = 115.00\,\text{V}\) is the voltage, and \(P\) is the power. So, we have: \[R = \frac{\left(115.00\,\text{V}\right)^2}{\frac{\left(0.4\,\text{kg}\right)\left(4180\,\frac{\text{J}}{\text{kg}\cdot\text{K}}\right)\left(75.00\,\text{K}\right)}{120\,\text{s}}}.\] Now, calculate the value of \(R\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Heat Transfer
Heat transfer is the process of thermal energy moving from one object or substance to another. It plays a crucial role in various applications, including heating water, which is common in our daily lives. When a physics student decides to heat water using an immersion heater, the concept of heat transfer is the primary phenomenon at work.

There are three main types of heat transfer: conduction, convection, and radiation. In the case of heating water, convection is the most relevant, as it involves the movement of heated fluid (water in this case) from one place to another, which distributes heat throughout the container.

An understanding of heat transfer is not only essential for physics students but is also valuable in everyday applications such as cooking, industrial processes, and even climate control systems like heaters and air conditioners.
Specific Heat Capacity
The specific heat capacity, often denoted as 'c', is a property that describes how much energy it takes to raise the temperature of a certain mass of a substance by one degree Celsius. It is measured in joules per kilogram per degree Kelvin (J/kg·K). Water, for example, has a high specific heat capacity of approximately 4180 J/kg·K, which means it requires a lot of energy to change its temperature.

This property is paramount in solving problems related to temperature changes. In our exercise, we used the specific heat capacity of water to calculate how much energy the immersion heater needed to raise the water temperature from 25.00°C to 100.00°C. The understanding of specific heat capacity helps students and engineers design efficient cooling and heating systems.
Electrical Power in Circuits
Electrical power quantifies the rate at which electrical energy is converted into another form of energy, such as heat, light, or motion. The SI unit for power is the watt (W), and one watt is equivalent to one joule per second (J/s).

In the context of the exercise, the electrical power used by the immersion heater is responsible for converting electrical energy into thermal energy to heat the water. By calculating the total energy required to achieve the desired temperature change and dividing it by the time it took, we find the average power consumption of the heater.

Understanding how to calculate electrical power is essential in various fields, from electrical engineering to household energy management, making it possible to optimize energy usage and decrease costs.
Ohm's Law and Electrical Resistance
Ohm's Law is a fundamental principle in the field of electrical engineering and physics. It states that the current 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 formula is given by V = IR, where V is the voltage, I is the current, and R is the resistance.

In the heater exercise, we reverse the logic of Ohm's Law to solve for resistance (R) using the formula R = V^2/P, given the voltage (V) and the power (P). Understanding Ohm's Law is not only critical for solving physics problems but is also crucial for anyone working with electrical circuits, as it helps predict how changing one quantity affects the others.

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Most popular questions from this chapter

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