91Ó°ÊÓ

Energy transfer is the process by which energy moves from one place to another or from one object to another. In this exercise, energy is being transferred from an electric heater to the water and the coffee mug. The energy transfer occurs in the form of heat. This heat energy then raises the temperature of both objects, ultimately bringing the water to a boil.

The ability of an object to absorb heat energy without significantly increasing in temperature varies between materials. This is known as the specific heat capacity. For example, water has a high specific heat capacity, which means it can absorb a lot of heat before its temperature rises significantly. The coffee mug, which is made from a different material, has its own specific heat capacity. In our exercise, the specific heat capacities are given as 4186 J/kg°C for water and 920 J/kg°C for the coffee mug.

Understanding this energy transfer is critical because it enables us to calculate the amount of energy required to achieve the desired temperature change.

Thermal energy is a kind of potential energy stored within a substance due to the motion and actions of its particles. When we heat an object, we increase its thermal energy, which in turn raises its temperature. In our example, the coffee mug and the water both undergo an increase in thermal energy as they are being heated.

The concept of temperature change (\( \Delta T \), which in this case is 85°C) is crucial as it tells us how much the thermal energy stored in the water and mug needs to increase. From this change, we can calculate how much heat energy (Q) is needed for each component, as seen in the calculations: For water:\[ Q_{\text{water}} = m_{\text{water}} \cdot c_{\text{water}} \cdot \Delta T \], and for the mug: \[ Q_{\text{mug}} = m_{\text{mug}} \cdot c_{\text{mug}} \cdot \Delta T \].

By calculating these energies separately, and then adding them, we get the total thermal energy input required to bring the water to a boil. The resulting value, 116310.5 J, represents the cumulative thermal energy increase for the system.

Having understood the energy transfer and thermal energy concepts, we are now equipped to calculate the power needed from the heater. Power, in physics, is defined as the rate at which energy is transferred or converted. It is given by the formula:

\[ P = \frac{Q_{\text{total}}}{t} \]

In this scenario, since we know that the total energy required is 116310.5 J and the heating time is 180 seconds, we can find the power of the heater by plugging these values into our formula:

\[ P = \frac{116310.5}{180} \approx 646.17 \text{ W} \]

This result indicates that the heater must have a minimum power rating of about 646.17 watts to achieve the desired outcome. Power calculation is fundamental in solving practical problems involving heating and energy transfer, ensuring that we choose the right equipment for the task at hand.