Animated Solution for Physics - Dual Nature of Matter and Radiation: A $2\text{ mW}$ laser operates at a wavelength of $500\text{ nm}$. The number of photons that will be emitted per second is [Given, Planck's constant $h=6.6\times 10^{-34}\text{ Js}$, speed of light $c=3.0\times 10^8\text{ m/s}$]

Imagine you are holding a standard laser pointer. When you press the button, a continuous, bright beam of light shoots out. To our eyes, it looks like a solid, unbroken stream. However, quantum mechanics tells us a different, much more fascinating story. That beam is actually a torrential downpour of tiny, indivisible packets of energy called photons.

In this problem, we are tasked with counting these invisible packets. We need to find out exactly how many photons are streaming out of a $2\text{ mW}$ laser every single second.

To solve this, we need to connect the macroscopic world (the total power of the laser) with the microscopic world (the energy of a single photon).

First, let's look at the macroscopic side. Power ($P$) is defined as the total energy emitted per unit of time.

If we let $n$ represent the number of photons emitted in one second, then the total energy emitted in that second is simply $n$ multiplied by the energy of one single photon ($E$).

Now, let's zoom into the microscopic side. How much energy does one photon carry? According to the Planck-Einstein relation, the energy of a photon is directly tied to its wavelength ($\lambda$).

Here, $h$ is Planck's constant ($6.6\times 10^{-34}\text{ Js}$) and $c$ is the speed of light ($3.0\times 10^8\text{ m/s}$).

By substituting this microscopic energy equation into our macroscopic power equation, we get our master formula:

We want to find $n$, the number of photons per second. Let's rearrange our master formula to isolate $n$:

Before we plug in the numbers, we must be incredibly careful with our units. This is where many students make a fatal error. The power is given in milliwatts ($\text{mW}$), which must be converted to Watts ($\text{W}$). The wavelength is in nanometers ($\text{nm}$), which must be converted to meters ($\text{m}$).

- $P=2\text{ mW}=2\times 10^{-3}\text{ W}$ - $\lambda =500\text{ nm}=500\times 10^{-9}\text{ m}$

Now, let's substitute these values into our rearranged equation:

Let's simplify the numerator and denominator separately to avoid confusion.

In the numerator: $2\times 500=1000$. Combining the powers of $10$, we get $10^{-3}\times 10^{-9}=10^{-12}$. So the numerator is $1000\times 10^{-12}$, which simplifies to $10^{-9}$.

In the denominator: $6.6\times 3.0=19.8$. Combining the powers of $10$, we get $10^{-34}\times 10^8=10^{-26}$. So the denominator is $19.8\times 10^{-26}$.

Putting it back together:

Calculating the fraction, $\frac{1}{19.8}\approx 0.0505$.

To match the format of our options, we can shift the decimal point two places to the right, which decreases the exponent by two:

Looking at our choices, the closest value is $5\times 10^{15}$. Therefore, option (b) is the correct answer. It's mind-blowing to think that a simple laser pointer emits over 5 quadrillion photons every single second!