What is the energy content per photon \((J)\) for light of frequency \(4.2 \times 10^{14}\) ? (a) \(2.8 \times 10^{-21}\) (b) \(2.5 \times 10^{-19}\) (c) \(2.8 \times 10^{-19}\) (d) \(2.5 \times 10^{-18}\)

Whenever we dive into the realm of quantum physics, Planck's equation becomes a central pillar. This equation, elegantly simple, connects the dots between the energy of a photon and the frequency of light it represents. In its essence, Planck's equation is expressed as:
\[ E = h \cdot f \]
where \( E \) denotes the energy of an individual photon, \( h \) represents Planck's constant, and \( f \) stands for the frequency of the light wave. This relationship illustrates the quantized nature of light, showing us that energy is not continuous but comes in discrete packets, or 'quanta' of energy. It's a crucial concept that paved the way for quantum mechanics and forever changed our understanding of light and energy.

The frequency of light is fundamental to understanding the behavior of electromagnetic waves, including visible light. Measured in hertz (Hz), frequency tells us how many waves pass a given point every second. Higher frequencies mean more waves per second, and according to Planck's equation, more energy per photon.

For instance, ultraviolet light has a higher frequency than visible light, thereby packing each of its photons with more energy. This concept is evident when considering the effects of UV light on the skin versus visible light—it has enough energy to cause sunburns! When we discuss light frequency in exercises like the one given, we're delving into this intrinsic property that dictates so much of the light's energy and its interactions with matter.

Planck's constant, symbolized by \( h \), is a fundamental constant in physics that has far-reaching implications in the quantum world. As part of Planck's equation, this constant is the proportionality factor between the energy of a photon and its frequency. With a value of approximately \( 6.62607015 \times 10^{-34} \text{ J} \cdot \text{s} \), Planck's constant is tiny, reminding us of the minuscule scale at which quantum phenomena occur.

Planck's constant is not just a number; it's a bridge between the macroscopic and microscopic worlds, and understanding its role is vital when calculating the energy content per photon, as highlighted in the provided exercise.

In the world of measurements and calculations, accuracy and precision are paramount. That's where significant figures come into play. They represent the digits in a number that contribute to its precision. When you perform calculations, like those using Planck's equation, it's important to round your final answer to the correct number of significant figures to reflect the precision of your measurement.

For example, the step-by-step solution provided used significant figures to round the calculated energy content per photon to \( 2.8 \times 10^{-19} \text{ J} \), aligning with the given options. This practice avoids overestimation of the precision of the result and ensures that the reported value honestly represents the input's accuracy. Remember, it's not just about getting the right answer—it's about conveying the right degree of certainty in your findings.