To address your question about the two identical metal balls at different temperatures, we need to delve into the principles of thermal radiation and the Stefan-Boltzmann law. Yes, the temperatures of the two balls will eventually equalize due to radiation, but let's break down how this process works and whether the rate of heat gained by the colder sphere aligns with the expectations from Stefan's law.
The Process of Thermal Radiation
When two objects are at different temperatures, heat transfer occurs from the hotter object to the cooler one until thermal equilibrium is reached. In this case, the ball at 600 K will radiate energy towards the ball at 300 K. This transfer happens through electromagnetic radiation, which can occur even in a vacuum, as there is no need for a medium like air or water.
Understanding Stefan-Boltzmann Law
The Stefan-Boltzmann law states that the power radiated by a black body per unit area is proportional to the fourth power of its absolute temperature. Mathematically, it can be expressed as:
where:
- P is the power radiated,
- σ is the Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K⁴),
- A is the surface area of the object, and
- T is the absolute temperature in Kelvin.
Heat Transfer Between the Balls
In your scenario, the rate of heat transfer from the hotter ball (T2 = 600 K) to the colder ball (T1 = 300 K) can be calculated using the difference in their radiative power. The power radiated by each ball can be expressed as:
- P1 = σA(T1^4) for the colder ball,
- P2 = σA(T2^4) for the hotter ball.
However, since the balls are identical and only the temperature differs, we focus on the net power transfer from the hotter to the cooler ball:
- Net Power = P2 - P1 = σA(T2^4 - T1^4).
Proportionality to Temperature Differences
Now, regarding your question about whether the rate of heat gained by the colder sphere is proportional to T2^4 - T1^4, the answer is yes. The net power transfer indeed follows this relationship. As the colder ball absorbs heat, its temperature will rise, and the rate of heat gain will change accordingly. However, the relationship is not linear; it is based on the fourth power of the temperatures.
Implications of Temperature Change
As the colder ball heats up, the temperature difference between the two balls decreases, which will reduce the rate of heat transfer over time. Eventually, when both balls reach the same temperature, the net power transfer will equal zero, and thermal equilibrium will be established.
Summary of Key Points
- Heat will transfer from the hotter ball to the cooler ball through radiation.
- The rate of heat gained by the colder ball is proportional to T2^4 - T1^4.
- As temperatures equalize, the rate of heat transfer decreases until equilibrium is reached.
In conclusion, the two metal balls will indeed equalize their temperatures through radiation, and the rate of heat gained by the colder sphere will follow the principles outlined by Stefan's law, demonstrating the fascinating interplay of thermal dynamics in a vacuum.
