Problem 2 - University Physics with Modern Physics

Question

BIO Response of the Eve. The human eve is most sensitive to green light of wavelength 505 \(\mathrm{nm}\) . Experiments have found that when people are kept in a dark room until their eyes adapt to the darkness, a single photon of green light will trigger receptor cells in the rods of the retina. (a) What is the frequency of this photon? (b) How much energy (in joules and electron volts) does it deliver to the receptor cells? (c) To appreciate what a smount of energy this is, calculate how fast a typical bacterium of mass \(9.5 \times 10^{-12}\) g would move if it had that much energy.

Step-by-Step Solution

Answer

(a) 5.94 x 10^14 Hz; (b) 3.94 x 10^-19 J, 2.46 eV; (c) 9.1 m/s.

Step 1: Calculate Frequency

To find the frequency of the photon, we use the equation \( c = \lambda \cdot f \), where \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), \( \lambda \) is the wavelength \( 505 \text{ nm} = 505 \times 10^{-9} \text{ m} \), and \( f \) is the frequency. Rearranging for frequency, \( f = \frac{c}{\lambda} \). Substituting the values, we get \( f = \frac{3.00 \times 10^8 \text{ m/s}}{505 \times 10^{-9} \text{ m}} = 5.94 \times 10^{14} \text{ Hz} \).

Step 2: Calculate Energy in Joules

The energy of a photon can be calculated using Planck's equation \( E = h \cdot f \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J·s} \), and \( f \) is the frequency \( 5.94 \times 10^{14} \text{ Hz} \). Thus, \( E = 6.626 \times 10^{-34} \cdot 5.94 \times 10^{14} = 3.94 \times 10^{-19} \text{ J} \).

Step 3: Convert Energy to Electron Volts

We convert the energy from joules to electron volts using the conversion factor: \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \). Therefore, \( E = \frac{3.94 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} = 2.46 \text{ eV} \).

Step 4: Calculate Bacterium Speed

To find the speed \( v \) of the bacterium, we use the equation for kinetic energy \( E = \frac{1}{2} mv^2 \), where \( E = 3.94 \times 10^{-19} \text{ J} \) and \( m = 9.5 \times 10^{-12} \text{ g} = 9.5 \times 10^{-15} \text{ kg} \). Rearranging for \( v \), we have \( v = \sqrt{\frac{2E}{m}} \). Substituting the values, \( v = \sqrt{\frac{2 \times 3.94 \times 10^{-19}}{9.5 \times 10^{-15}}} = 9.1 \text{ m/s} \).

Key Concepts

Wavelength

Wavelength is a fundamental concept in the study of light and other electromagnetic waves. It represents the distance between two successive peaks or troughs of a wave. This property is typically measured in nanometers (nm) when discussing light, with one nanometer being equal to one billionth of a meter. For example, the green light that human eyes are most sensitive to has a wavelength of 505 nm.
Understanding wavelength helps in determining other characteristics of a wave, such as its frequency and energy. This relationship is crucial in physics, where phenomena such as the color of light, its interaction with materials, and its transmission through different mediums are analyzed.
In the biological context, the sensitivity of the human eye to different wavelengths allows us to perceive colors. This sensitivity peaks in areas where the wavelength of light corresponds to the green spectrum, which helps us see a variety of colors under different lighting conditions.

Frequency

Frequency refers to the number of wave cycles that pass a given point per unit of time. It is an essential property of waves, measured in hertz (Hz), where one hertz equates to one cycle per second. In the context of electromagnetic waves, such as light, frequency is inversely related to wavelength.
The equation of light speed, given by \( c = \lambda \cdot f \), where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( f \) is the frequency, reveals that as wavelength increases, frequency decreases, and vice versa. By rearranging this equation, we can calculate the frequency if the wavelength is known and vice versa.
In our example, a green photon with a wavelength of 505 nm has a corresponding frequency. This helps explain why different colors of light have unique properties, as each frequency corresponds to specific energy levels and interactions with matter.

Planck's Constant

Planck's constant is a vital quantity in quantum mechanics that bridges the properties of particles with wave-like behavior to light's energy. Its value is approximately \( 6.626 \times 10^{-34} \text{ J·s} \).
This constant is used to calculate the energy of a photon, the basic unit of light, through the equation \( E = h \cdot f \), where \( E \) is energy, \( h \) is Planck's constant, and \( f \) is frequency. Essentially, it shows that a photon's energy is directly proportional to its frequency.
Understanding Planck’s constant is crucial not just in calculating photon energy but also in explaining phenomena like the photoelectric effect, where light can eject electrons from materials, and the foundational aspects of quantum theory.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. For calculating kinetic energy from the context of a photon’s energy transferred to a particle, we use the formula \( E = \frac{1}{2} mv^2 \), where \( E \) represents energy, \( m \) is mass, and \( v \) is velocity.
This relationship allows us to determine how fast a particle will move if it absorbs a specific amount of energy. For instance, in the original exercise, we see this applied to calculate the speed of a bacterium when exposed to the energy of a single photon.
Kinetic energy is a core principle in physics applied to both macroscopic and microscopic scales, from driving vehicles to explaining atomic behaviors when they interact with electromagnetic radiation.