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Specific heat capacity is a property of a material that indicates how much heat energy is needed to raise the temperature of a unit mass of the substance by one degree Celsius. For the glass plate in this exercise, the specific heat capacity is given as 840 \( \mathrm{J} / (\mathrm{kg} \cdot \mathrm{C}^{\circ}) \). This means that 840 joules of energy are required to increase the temperature of 1 kilogram of glass by 1 degree Celsius.
Understanding specific heat capacity is crucial for solving problems involving temperature changes. It helps to determine the quantity of heat energy needed to achieve the desired change.
Given a mass \( m \) and a temperature change \( \Delta T \), the total heat \( Q \) needed can be calculated using the formula:

• \( Q = m \cdot c \cdot \Delta T \)

Here, \( c \) represents the specific heat capacity. In our problem, using the given values resulted in \( Q = 840 \) Joules, indicating the heat energy needed to raise the glass plate's temperature by 2 degrees Celsius.

Photons are tiny packets of energy that travel at the speed of light. The energy of a photon depends on its wavelength—the shorter the wavelength, the higher the energy. To find how many photons are needed to raise the temperature of the glass plate, we need to understand the energy each type of photon carries.
The energy of a photon \( E \) can be calculated using the equation:

• \( E = \frac{hc}{\lambda} \)

Here, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}) \), \( c \) is the speed of light \( (3 \times 10^{8} \mathrm{m/s}) \), and \( \lambda \) is the wavelength of the photon.
Photon energy is vital in this context because it allows us to determine how many photons are needed to provide the required thermal energy. By taking the heat requirement from the glass plate formula and dividing it by the energy per photon, we obtain the number of photons needed.

Wavelength is a critical feature of light that determines its energy. Light can be categorized into different types, such as infrared and blue light, based on its wavelength.
In this exercise, the infrared light has a wavelength of \( 6.0 \times 10^{-5} \) meters, while blue light has a shorter wavelength of \( 4.7 \times 10^{-7} \) meters. The shorter wavelength of blue light indicates it has more energy per photon compared to infrared light.
Wavelength is directly related to photon energy by the photon equation:

• \( E = \frac{hc}{\lambda} \)

Thus, knowing the wavelength helps us figure out how much energy each photon carries, which is crucial for calculating how many photons are needed to achieve a certain amount of energy, like raising the glass plate temperature in our case.

Temperature change refers to the difference in temperature that is being targeted after the glass absorbs the energy provided by photons.
In our given problem, the target temperature change \( \Delta T \) is 2 degrees Celsius. Temperature change is a driving factor for calculating the amount of heat required, as it appears in the heat calculation formula:

• \( Q = m \cdot c \cdot \Delta T \)

Without specifying the temperature change, it wouldn't be possible to calculate the energy requirement to warm the glass.
Understanding the relationship between the heat provided by the photons and the resulting temperature change is fundamental to solving real-world thermodynamics problems. This knowledge enables us to predict how much energy is needed and, consequently, how many photons are required to achieve the desired thermal effect.