Question

The eyes of a certain reptile family pass a single visual signal to a brain when the visual receptors are struck by photons of wavelength $662nm$. If a total energy of $3 \times {10^{ - 14}}J$ is required to trip the signal, what is the minimum number of photons that must strike the receptor.$(h = 6.62 \times {10^{ - 34}}Js)$.

Answer 1

First we will write all the data given in the question. Then we will write the speed of light. Now we will have to calculate the frequency of the photon. From frequency we can calculate the energy of 1 proton. From that we can calculate the number of protons required by dividing the total energy by the energy of one photon. From there we will get our final answer.

Complete step-by-step solution: Step1. The data given in the question is:
The wavelength from which the reptile is struck: $662nm \Rightarrow 662 \times {10^{ - 9}}\;m$
The total energy required to trip the signal: $3 \times {10^{ - 14}}J$
The plants constant $h = 6.62 \times {10^{ - 34}}Js$
We need to find the minimum number of photons.
Step2. The speed of light is $3 \times {10^8}m/s$. It is represented by ‘c’
Step3. Now we can calculate the frequency of photons. The frequency is the speed of light divided by the wavelength.
$frequency(\nu ) = \dfrac{c}{\lambda }$ , $\lambda$ is the wavelength.
$\nu = \dfrac{{3 \times {{10}^8}}}{{662 \times {{10}^{ - 9}}}}$
$\Rightarrow \nu = \dfrac{{3 \times {{10}^{17}}}}{{662}} = 4.53 \times {10^{14}}{s^{ - 1}}$
We got the frequency here.
Step4. Now we calculate the energy of one photon. It is the plank constant multiplied to the frequency.
$E = h\nu$
$\Rightarrow E = 6.62 \times {10^{ - 34}} \times 4.53 \times {10^{14}}$
$\Rightarrow E = 2.9988 \times {10^{ - 19}}J$
Step5. Now we can find the protons required by dividing the total energy given by energy of one photon.
$protons = \dfrac{{{E_{total}}}}{{{E_{1proton}}}}$
$\Rightarrow protons = \dfrac{{3.0 \times {{10}^{ - 14}}}}{{2.9988 \times {{10}^{ - 19}}}} = {10^5}$
The total protons needed are ${10^5}$ that must strike the receptor.

Note: The frequency is the number of occurrences of repeating an event per unit of time. The unit of frequency is hertz. It can be defined as the one occurrence of a event per second. It is often denoted by $\nu$.