Animated Solution for Physics - Electromagnetic Waves: Suppose that intensity of a laser is $\left(\frac{315}{\pi }\right){\text{ W/m}}^2$. The rms electric field (in V/m) associated with this source is close to the nearest integer is ..... . (Take, ${\epsilon }_0=8.86\times 10^{-12}{\text{ C}}^2{\text{Nm}}^{-2}$ and $c=3\times 10^8{\text{ ms}}^{-1}$)

Imagine standing in a dark room, and suddenly, a brilliant, focused beam of red light pierces through the darkness. That’s a laser. But what exactly is that beam made of? At its core, a laser is an electromagnetic wave—a synchronized dance of electric and magnetic fields oscillating through space at the cosmic speed limit: the speed of light.

When we talk about the 'intensity' of this laser, we are talking about power. Specifically, it is the amount of energy that crosses a unit area every single second. If you were to hold your hand in front of the beam, the intensity dictates how much heat you would feel. In this problem, we are given a very specific intensity: $I=\frac{315}{\pi }{\text{ W/m}}^2$.

Our mission is to look past the macroscopic brightness of the beam and calculate the microscopic strength of the electric field that is oscillating within it. Specifically, we need to find the Root Mean Square (RMS) electric field. Let's embark on this journey.

To bridge the gap between the macroscopic intensity and the microscopic electric field, we need to rely on the fundamental principles of electromagnetism formulated by James Clerk Maxwell.

The energy in an electromagnetic wave is shared equally between the electric field and the magnetic field. The average energy density (energy per unit volume) stored in the electric field is $\frac{1}{4}{\epsilon }_0{E}_0^2$, and the same amount is stored in the magnetic field. Therefore, the total average energy density is $\frac{1}{2}{\epsilon }_0{E}_0^2$.

Intensity is simply this energy density moving at the speed of light, $c$. This gives us our master equation: Here, $E_0$ represents the peak amplitude of the electric field—the absolute maximum value it reaches during its oscillation.

However, the question throws a slight curveball. It doesn't ask for the peak electric field; it asks for the RMS (Root Mean Square) electric field, $E_{rms}$.

Why do physicists love RMS values? Because an electromagnetic wave is sinusoidal—it constantly fluctuates between positive and negative values, averaging out to zero. The RMS value gives us a meaningful 'effective' value, much like how we measure AC voltage in our homes. For any sinusoidal wave, the RMS value is the peak value divided by the square root of two: If we square both sides of this relation, we get a very useful substitution:

Now, let's bring this substitution back into our master equation. This is where the mathematical elegance of physics shines through. By substituting $E_0^2$ with $2E_{rms}^2$, we get:

Notice what happens here? The fraction $\frac{1}{2}$ and the integer $2$ cancel each other out perfectly. We are left with a beautifully streamlined equation that directly connects intensity to the RMS electric field:

This equation is powerful. It tells us that the intensity is directly proportional to the square of the RMS electric field.

Our goal is to find $E_{rms}$, so let's isolate it. First, we divide both sides by the product of the permittivity of free space (${\epsilon }_0$) and the speed of light ($c$):

Finally, to strip away the square, we take the square root of both sides: We now have our final algebraic weapon. All that remains is the numerical execution.

This is the phase where many students make silly mistakes. We have a formula, and we have the numbers. Let's proceed with caution and precision.

We are given: - $I=\frac{315}{\pi }{\text{ W/m}}^2$ - ${\epsilon }_0=8.86\times 10^{-12}{\text{ C}}^2{\text{Nm}}^{-2}$ - $c=3\times 10^8{\text{ ms}}^{-1}$

Let's plug these into our derived formula:

When faced with a complex fraction like this, the best strategy is to conquer it in pieces. Let's start with the denominator. We need to multiply the constants ${\epsilon }_0$ and $c$: First, multiply the significant digits: $8.86\times 3=26.58$. Next, combine the powers of ten using the laws of exponents: $10^{-12}\times 10^8=10^{-4}$. So, our denominator simplifies to:

Now, let's look at the numerator. We have $\frac{315}{\pi }$. Using the standard approximation for pi ($\pi \approx 3.14159$), we can evaluate this:

Now, let's reconstruct our square root with these simplified pieces:

To make the division easier, let's move the $10^{-4}$ from the denominator to the numerator, which changes its sign to $10^4$:

Performing the division gives us:

We are at the final step. We need to find the square root of $37722.7$. If you don't have a calculator, you can estimate this by knowing your squares. We know that $190^2=36100$ and $200^2=40000$. Our number is somewhere in between, closer to $190$. Calculating it precisely yields:

The problem asks for the value close to the nearest integer. Looking at $194.22$, the decimal part is less than $0.5$, so we round down.

Our final, triumphant answer is 194. You have successfully navigated from the macroscopic intensity of a laser beam down to the exact numerical value of its oscillating electric field!