Question

Consider X-rays \(\left(\lambda \sim 10^{-10} \mathrm{~m}\right)\) and green light \(\left(\lambda \sim 10^{-7} \mathrm{~m}\right)\) (a) Do a calculation to show how much more energetic X-rays are than green light. (b) Why are X-rays dangerous whereas green light is not?

Short answer

X-rays are about 1000 times more energetic than green light, making them more likely to interact with and damage molecules in the body, such as DNA, potentially leading to harmful effects like cancer. In contrast, green light has lower energy and is less likely to cause cellular damage, making it safe for everyday exposure.

Step-by-step solution

Part (a): Calculate the energy of X-rays and green light

In order to calculate the energies of X-rays and green light, we can use the energy-wavelength equation for photons: \[E = \dfrac{hc}{\lambda}\] where - E is the energy of the photon - h is Planck's constant (\(6.63 \times 10^{-34}\) Js) - c is the speed of light (\(3.00 \times 10^{8}\) m/s) - λ is the wavelength of the photon First, compute the energies for both X-rays and green light: For X-rays, \(\lambda_{xray} \sim 10^{-10}\mathrm{~m}\) \[E_{xray} = \dfrac{(6.63 \times 10^{-34}\mathrm{~Js})(3.00 \times 10^{8}\mathrm{~m/s})}{10^{-10}\mathrm{~m}}\] \[E_{xray} \approx 1.99 \times 10^{-16} \mathrm{J}\] For green light, \(\lambda_{green} \sim 10^{-7}\mathrm{~m}\) \[E_{green} = \dfrac{(6.63 \times 10^{-34}\mathrm{~Js})(3.00 \times 10^{8}\mathrm{~m/s})}{10^{-7}\mathrm{~m}}\] \[E_{green} \approx 1.99 \times 10^{-19} \mathrm{J}\] Now, calculate the ratio of energies: \[ratio = \dfrac{E_{xray}}{E_{green}} = \dfrac{1.99 \times 10^{-16} \mathrm{J}}{1.99 \times 10^{-19} \mathrm{J}}\] \[ratio \approx 1000\]

Part (b): Reasons for dangers of X-rays vs. safety of green light

X-rays are more dangerous than green light because of their higher energy. The ratio we calculated in part (a) shows that X-rays are about 1000 times more energetic than green light. Higher-energy photons are more likely to interact with and damage molecules in the body, such as DNA. These interactions can lead to harmful effects, such as the development of cancer. Green light, on the other hand, has lower energy and is less likely to cause damage to our cells and, therefore, is considered safe for everyday exposure.

Key concepts

X-rays

X-rays are a form of electromagnetic radiation, just like visible light, but they have some distinct traits that set them apart in the electromagnetic spectrum. Essentially, X-rays have much shorter wavelengths, which translates into higher energy levels. They typically have wavelengths in the range of 0.01 to 10 nanometers (nm), significantly smaller than visible light.

Due to their high energy, X-rays can penetrate various materials, including the human body, which makes them incredibly valuable in medical imaging techniques such as radiography. However, it is this same penetrating power that renders them potentially harmful; they can damage biological tissues and cause mutations in DNA, leading to increased risks of cancer. That's why precautions, such as wearing lead aprons, are taken when using X-rays in medical settings to minimize exposure.

Green Light

Green light, with a wavelength of about 500 nanometers (nm), sits right in the middle of the visible light spectrum. These wavelengths resonate with how our eyes are built to sense light, so we find green light particularly intense and bright. Unlike X-rays, the energy of green light photons is much lower, putting them in a range that is safe for our cells and tissues.

Green light is commonly experienced every day; it's the light we see all around us in nature. Plants predominantly use it for photosynthesis to convert sunlight into chemical energy. Additionally, because of its lower energy level compared to X-rays, green light is not capable of breaking molecular bonds and is therefore not harmful to our bodies, making it safe for both daily exposure and various applications, such as traffic signals and lighting.

Planck's Constant

Planck's constant is a crucial value in quantum mechanics. It is denoted by the symbol 'h' and has an approximate value of 6.63 x 10^-34 Joule seconds (Js). This constant is used to describe the size of the discrete units of energy, or quanta, involved in interactions, such as the emission or absorption of light by atoms.

Significance in Photon Energy

Planck's constant directly relates to the energy of photons, the fundamental particles of light, through the relationship encapsulated by the formula:
\(E = hf\), where 'E' is energy and 'f' is frequency. The constant ensures that as frequency increases, so does the energy. It enables us to calculate the precise quanta of energy for photons of specific wavelengths and frequencies, thus playing a monumental role in our understanding of the universe.

Wavelength-Energy Equation

The wavelength-energy equation is a pivotal concept in physics, highlighting the inverse relationship between the wavelength of light and its energy. The equation is described as:
\[E = \frac{hc}{\lambda}\]
where 'E' represents the energy of a photon, 'h' is Planck's constant, 'c' is the speed of light, and ‘λ’ denotes the wavelength.

What is profound about this formula is that it encapsulates how every light wave, from the very low-energy radio waves to the extremely high-energy gamma rays, carries quantized energy. This energy can have significant effects, from merely illuminating a room with visible light to initiating chemical reactions or causing ionization in biological tissues, as is the case with X-rays. Understanding the equation aids in grasping why different types of light interact with matter in such varied ways.