A 200-W heater emits a 1.5-\mum radiation. (a) What value of the energy quantum does it emit? (b) Assuming that the specific heat of a 4.0-kg body is 0.83kcal/kg \cdot K, how many of these photons must be absorbed by the body to increase its temperature by 2 K? (c) How long does the heating process in (b) take, assuming that all radiation emitted by the heater gets absorbed by the body?

Understanding Planck's equation is essential for students to identify how the energy of photons relates to the frequency of electromagnetic radiation. According to Max Planck's theory, energy is not continuous, but comes in discrete units called quanta.

Planck's equation is symbolically expressed as E = hf, where E is the energy of a single quantum (or photon), h is Planck's constant (6.63 x 10^-34 Joule seconds), and f is the frequency of the radiation.

Why is Planck's equation pivotal?

It interlinks the concept of quantization with the measurable property of light—frequency—thereby allowing us to calculate the specific energy of photons emitted by any radiation source, including our 200-W heater from the exercise. Multiply Planck's constant with the emitted radiation frequency, and you obtain the energy for each photon. For students, this aids in visualizing energy transfer on a quantum scale, which is a cornerstone of modern physics.

In our exercise, we delve into the specific heat of a substance, which is a measure of the amount of heat required to change the temperature of a unit mass of the substance by one degree Celsius or Kelvin (the numeric values for temperature changes are the same in both scales). The specific heat equation is given by Q = mcΔT, where Q is the heat absorbed or released, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.

An important note for students is that different substances have different capacities for storing thermal energy, hence the variation in specific heat values. For instance, the specific heat of water is high at 1 cal/g·K, allowing it to absorb lots of heat without significantly changing in temperature—a fact that plays a crucial role in Earth's climate and ecosystems. In contrast, metals like copper have lower specific heats, making them excellent for cooking utensils due to their fast heating and cooling properties.

Practical Application

Calculating the increase in temperature for our 4.0-kg body, given its specific heat, underscores not just a purely academic exercise, but the same scientific principles used to design climate control systems and energy-efficient buildings.

Lastly, photon energy calculation is integral to understanding the energy transfer from radiation to matter. This calculation uses the previously discussed Planck's equation in combination with the energy required to change the temperature of an object. In our exercise, we establish the total energy (Q), and then determine how many photons (N) with a certain energy (E) must be absorbed to achieve the desired temperature increase.

To perform this calculation, we divide the total energy by the energy per photon: N = Q / E. This provides a direct link to the concept of quantization as it relates to heat transfer. Such calculations are not only relevant in theoretical physics but also in practical applications such as medical therapies using light, solar panel technologies, and even culinary techniques involving infrared cooking.

Real-World Implications

For students, grasping photon energy calculation is also about understanding the significance of energy density in different frequencies of light, implications for harnessing solar energy, and even the workings of laser technology.