Animated Solution for Physics - Electromagnetic Waves: A red LED emits light at 0.1 W uniformly around it. The amplitude of the electric field of the light at a distance of 1 m from the diode is

When we look at a simple red LED, we see a tiny dot of light. But from a physics perspective, this tiny dot is a powerhouse radiating electromagnetic energy into the space around it. To understand how this energy behaves, we need to bridge the gap between the macroscopic world of 'Watts' and the microscopic world of 'Electric Fields'.

Imagine the LED as a perfect point source. It emits light uniformly in all directions. As the light travels outward, it forms expanding spherical wavefronts. The energy is spread thinner and thinner as the sphere grows larger.

Power and Area

First, let's define Intensity ($I$). Intensity is simply the amount of power crossing a unit area. Since our LED emits light uniformly, the power $P$ is distributed over the surface area of a sphere with radius $r$. The formula for the surface area of a sphere is $4\pi r^2$.

Therefore, the intensity at a distance $r$ is given by:

This tells us how much energy hits a specific patch of space every second.

Electromagnetic Waves

But light is an electromagnetic wave! It consists of oscillating electric and magnetic fields. The intensity of an electromagnetic wave is directly tied to the amplitude (the peak value) of its electric field, $E_0$. The relationship is given by the fundamental equation:

Here, ${ϵ}_0$ is the permittivity of free space, and $c$ is the speed of light. Notice the factor of $\frac{1}{2}$? This comes from averaging the rapidly oscillating ${\sin }^2(\omega t)$ term over a full cycle.

Now, we have two different ways to describe the exact same physical reality—the intensity of the light. By equating these two expressions, we can find the hidden electric field amplitude created by our macroscopic LED.

Our goal is to find $E_0$. Let's rearrange the equation to isolate $E_0^2$:

Now, we substitute the given values into our master equation. We know the power $P=0.1$ W, the distance $r=1$ m, and the speed of light $c=3\times 10^8$ m/s.

Crucially, remember the electrostatic constant: $\frac{1}{4\pi {ϵ}_0}=9\times 10^9$ N m${}^2$/C${}^2$. This makes our calculation much cleaner!

Simplifying the numerator gives $1.8\times 10^9$. Dividing this by $3\times 10^8$ yields a beautifully simple result:

Taking the square root, we find the amplitude of the electric field:

And there we have it! The electric field oscillating at a distance of 1 meter from this tiny LED has a peak strength of $2.45$ Volts per meter.