what is the instantaneous energy density ue(t) in the electric field of the wave? This is a topic that many people are looking for. amritsang.org is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, amritsang.org would like to introduce to you Energy Density of Electromagnetic Waves. Following along are instructions in the video below:
Lets continue our discussion on electromagnetic radiation. Also known as electromagnetic waves. Now just just like mechanical waves carry energy.
When they propagate through a medium electromagnetic waves. Also energy. When they propagate over vast distances through empty space now because electromagnetic waves are composed of alternating electric and magnetic fields that basically implies all the energy stored inside electromagnetic waves is stored within the electric and magnetic fields so lets suppose we have the following electromagnetic wave.
So we see that along the y axis. When the wave is traveling along the x axis. The electric field shown in green is moving is changing along the y axis.
While our magnetic field is shown in blue and is changing along the z axis. So the wave is propagating through our empty space. As its traveling in the positive direction along the x axis and all the energy is stored within these electric and magnetic fields now in our discussion on the energy stored inside waves.
It is very useful to discuss energy density rather than simply energy now recall that energy density is simply the quantity of energy stored per unit volume so that means the units of energy density are joules per meter cubed. Now in our discussion on the energy stored inside electric fields. We were able to show that the energy density inside electric field is equal to 1 2.
Multiplied by epsilon naught multiplied by the square of the electric field and in our discussion on the energy stored inside magnetic fields we were able to show that the energy density within magnetic fields is equal to 1 2 multiplied by the square of the magnetic field divided by mu naught. Where epsilon naught is the permittivity of free space and mew naught is the permeability of free space. These are both constants now because as we see in this depiction of our electromagnetic wave.
Because electromagnetic waves. Consists of both electric fields and magnetic fields that means the total energy density within our electromagnetic wave given by lowercase u. Is equal to the sum of the energy density of the electric field and the energy density of our magnetic field.
Now. You can be replaced with this equation and ub can be replaced with this equation. And we get the following result.
So this equation essentially gives us the total energy density stored inside our profit propagating electromagnetic wave. Now we can also use this equation to essentially determine the energy density of that propagating electromagnetic wave. Only in terms of the electric field.
So we want to determine the equation for the energy density in terms of only the electric field e. Now recall in a previous lecture. We were able to show that our speed of our electromagnetic wave in empty space c.
Is equal to the ratio of the electric field e. And the magnetic field b. And.
We said. That this is equal to 1 divided by the square root of the product of epsilon. Not the permittivity of free space and u.
Not the permeability of free space. Now if we take this equation this ratio is this ratio. We can use this to express our magnetic field in terms of the electric field and these two constants as shown in the following equation.
So the magnetic field b. Is equal to the product of the electric field e. And the square of the product of these.
Two constants now if we take this equation. And we replace our magnetic field b. With this product.
We get the following result now if we square the product of e and the square root of these two values. We get e squared multiplied by epsilon not multiplied by mu not the radical simply disappears now because u knot will appear on top and bottom. We can cancel that out and we get the following result so u.
The energy density of our electromagnetic wave is equal to this one half epsilon not multiplied by the square of e. One half e squared multiplied by epsilon not so if we combine these two terms. The twos will cancel and relax with the following equation.
So we see that the energy density of our electromagnetic wave is equal to the product of e squared and epsilon. Not the permeate ivities of free space. So this is the equation that gives us the total energy density of the electromagnetic wave.
Only in terms of the electric field now. If we follow a similar procedure. We can also determine the equation for the energy density only in terms of the magnetic field b.
So once again. We begin with this equation e divided by b. Is equal to 1 divided by the square root of u.
Naught. Multiplied by epsilon naught. So now instead of expressing b in terms of e.
We want to express e inter of b. And that is shown in this equation. So now once again.
We take this equation and we replace essentially our electric field with our magnetic field as shown in this result. So notice that this quantity. Squared simply amounts to b.
Squared. Divided by the radical disappears. Mu naught.
Multiplied by epsilon naught. So now we have epsilon naught on top and bottom. We can cancel those out and were left with the following result so the energy density of our electromagnetic wave is equal to the sum of these two terms.
So we sum up these two terms the twos cancel and we see that the energy density of our electromagnetic wave is equal to b. Squared. Divided by mu naught.
The permeability of free space. So we have three different equations that each give us the energy density of our electromagnetic wave equation. One describes our energy density in terms of these two constants and the electric field and magnetic field at any given moment in time the second equation gives us the energy density of the electromagnetic wave in terms of the permeate city of free space.
And the electric field and finally the final equation gives us the energy density in terms of the permeability of free space and our magnetic field b. So notice that e and b. Will present the strength of our fields at any given moment in time of our electromagnetic wave that is propagating through empty space.
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