Problem 61 - University Physics with Modern Physics

Question

A Nichrome heating element that has resistance 28.0\(\Omega\) is connected to a battery that has emf 96.0 \(\mathrm{V}\) and internal resistance 1.2\(\Omega\) . An aluminum cup with mass 0.130 kg contains 0.200 \(\mathrm{kg}\) of water. The heating element is placed in the water and the electrical energy dissipated in the resistance of the heating element all goes into the cup and water. The element itself has very small mass. How much time does it take for the temperature of the cup and water to rise from \(21.2^{\circ} \mathrm{C}\) to \(34.5^{\circ} \mathrm{C}\) ? (The change of the resistance of the Nichrome due to its temperature change can be neglected.)

Step-by-Step Solution

Answer

It takes approximately 48.0 seconds to heat the cup and water.

Step 1: Calculate Total Resistance in Circuit

First, find the total resistance in the circuit. The total resistance like any series circuit is the sum of the internal resistance of the battery and the resistance of the Nichrome wire. Thus, \( R_{total} = R_{nichrome} + R_{internal} = 28.0\,\Omega + 1.2\,\Omega = 29.2\,\Omega \).

Step 2: Calculate Current in the Circuit

Using Ohm's Law, calculate the current flowing through the circuit. Ohm's Law is \( V = IR \). Rearrange to find the current: \( I = \frac{V}{R} = \frac{96.0\,\text{V}}{29.2\,\Omega} \approx 3.29\,\text{A} \).

Step 3: Calculate Power Dissipated by the Heating Element

The power dissipated by the heating element is what is used to heat the water and the cup. Use the formula \( P = I^2R \), where \( R \) is the resistance of the Nichrome wire. This gives \( P = (3.29\,\text{A})^2 \times 28.0\,\Omega \approx 302.5\,\text{W} \).

Step 4: Determine Energy Required to Heat Water and Cup

Determine the energy required to raise the temperature of the entire system (cup and water) using the formula \( Q = mc\Delta T \). The specific heat of aluminum is \( 900\, \text{J/kg}^\circ\text{C} \) and for water is \( 4184\, \text{J/kg}^\circ\text{C} \). Thus, the total energy \( Q = (0.130\,\text{kg} \times 900\, \text{J/kg}^\circ\text{C} + 0.200\,\text{kg} \times 4184\, \text{J/kg}^\circ\text{C}) \times (34.5^\circ\text{C} - 21.2^\circ\text{C}) \approx 14509.3\,\text{J} \).

Step 5: Calculate Time Required to Heat Cup and Water

Using the power calculated earlier, calculate the time required. Time can be found from the formula \( t = \frac{Q}{P} \). Substitute the values \( t = \frac{14509.3\,\text{J}}{302.5\,\text{W}} \approx 48.0\,\text{s} \).

Key Concepts

Ohm's Law

Ohm's Law is a fundamental concept in physics, especially important in the study of electrical circuits. It is utilized to determine the relationship between voltage, current, and resistance in an electric circuit. Ohm's Law is expressed through the equation:

• \( V = IR \)

where:

• \( V \) is the voltage across the circuit (in volts),
• \( I \) is the current (in amperes),
• \( R \) is the resistance (in ohms).

This relationship illustrates how the current through a conductor is directly proportional to the voltage across it, with the resistance being the proportionality constant. In the exercise provided, Ohm’s Law helps in calculating the current flowing through the circuit by rearranging the formula to:

• \( I = \frac{V}{R} \)

using the known values of voltage and total resistance. This calculation is crucial as it informs further steps like determining power dissipation, which relates to the heating process of the water and the cup.

Specific Heat Capacity

Specific heat capacity is a vital concept in thermal physics. It refers to the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. The formula for specific heat capacity is:

• \( Q = mc\Delta T \)

where:

• \( Q \) is the heat energy transferred (in joules),
• \( m \) is the mass of the substance (in kilograms),
• \( c \) is the specific heat capacity (\( ext{J/kg}^ ext{°C} \)),
• \( \Delta T \) is the temperature change (in degrees Celsius).

In the context of the exercise, the specific heat capacities of both aluminum and water are utilized to determine the total energy required to increase the temperature from \( 21.2^ ext{°C} \) to \( 34.5^ ext{°C} \). Understanding the specific heat capacity of different materials is crucial because it influences how much energy is absorbed and consequently how much time it will take for the temperature to rise. Since water and aluminum have different specific heat capacities, they absorb heat at different rates, which needs to be taken into account when calculating the total energy required.

Electrical Circuits

An electrical circuit is a pathway in which electrons flow from a voltage or current source through connected components. A simple electrical circuit features essential components like a power source (battery), conductive path (wires), and a resistor which can be a heating element in practical scenarios. There are two main types of circuits:

• **Series Circuits:** Where components are connected end-to-end, so the same current flows through each component.
• **Parallel Circuits:** Where components are connected across common points or junctions, each component having its own path for the current.

In the provided problem, a series circuit is employed with a Nichrome heating element and battery. The total resistance in this series circuit is simply the sum of the individual resistances, namely of the Nichrome wire and the internal resistance of the battery. This total resistance allows calculating the current using Ohm’s Law. Knowing the current, we can further compute the power dissipated by the heating element, which is crucial for understanding energy conversion in the circuit. This energy conversion is ultimately responsible for heating the water and aluminum cup, showcasing a key application of electrical circuits in everyday devices.