 Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes this energy is obvious, such as in the warmth of the summer sun. And sometimes this energy has some practical purposes, such as using lasers for eye surgery. Other times it is more subtle, such as the unfelt energy of gamma rays which can destroy living cells. Electromagnetic waves carry energy from one region to another, just like mechanical waves. And these electromagnetic waves are made up of electric and magnetic fields. So both of these fields must be contributing to the total energy of the wave. To understand how to utilize this energy, it's helpful to derive detailed relationships for the energy in an electromagnetic wave. And that is what we will explore in this video. Alright, let's start off by recapping the field energies for the electric and magnetic field. Now for electric field, we can take a charged capacitor. And when it is charged, you will have an electric field going from one plate to the other. We know that the energy stored in the capacitor will be half Cv square, where C is the capacitance of this parallel plate capacitor and V is the potential difference across the plates. And because we want to figure out the electric field energy, we should have E in this expression. Turns out with some simple derivation, the energy density that is energy per unit volume comes out to be epsilon naught E square by 2. And there is a video where we actually derive this. And I strongly encourage you to go check that out if this feels a little fuzzy. Now similarly for magnetic field energy, we can take an inductor with a current flowing through it. And this is how the inductor looks like when we draw it in circuit diagrams. Now there will be a magnetic field produced as there is some current flowing in the coils. And the energy of that magnetic field is given by this expression, where L is the self-inductance of the solenoid of the inductor and I is the current flowing through it. This leads to the magnetic field density which is given by this expression, that is B square divided by 2 mu naught. And the energy per unit volume or the energy density for an electromagnetic wave must be the sum of these two energy densities. So in a region of space where E and B fields are present, the total energy density is given by the addition of electric field and magnetic field energy densities. You might wonder if these expressions also work for fields that are generated by let's say some radio station or even by the sun itself as that is the primary source of electromagnetic radiation. And turns out these expressions do work. They only care about what the field is at any one given single point and not its origin. These expressions hold true in general. Doesn't matter if we have capacitors, inductors or not. If we have electric field at a point, we can calculate its energy density by this expression as the same goes for magnetic field. For example, it doesn't matter if electric field is produced by a static charge or by a changing magnetic field. It is like saying that the kinetic energy of a moving ball is half MV square. Whether it got that energy by rolling down a hill or by a kick from you, it doesn't matter. At any one given point, the energy that it has is half MV square. We also know that light is a form of electromagnetic wave and it could have any source if there can be a light bulb or even a sun. And if I want to know how much energy light from the sun is carrying, all I need to know is how much energy is carried by the electric and magnetic fields. And it is absolutely insane that we can figure out the energy that light is carrying, something that is coming from the sun, just by analyzing capacitors and inductors. This would have been completely impossible a few centuries ago. But we can do this analysis now because now we know that light is a form of electromagnetic wave and the energy per unit volume that it might be carrying is just the sum of the energy densities of electric and magnetic fields. Since we have two fields contributing to the total energy in an electromagnetic wave, we can ask ourselves which field contributes more energy or which fields energy density is more. Why don't you think about it? Pause the video and see if you can come up with a way to figure that out. All right, I hope you have given it a thought. Now, we can compare these two densities. First, let me put in the expressions for the energy density. So, this is epsilon not E square by 2. And this is B square by 2 mu not. If let's say the electric field density was expressed entirely with the variable B or mu not, or if the magnetic field density was expressed entirely with the variable E or epsilon not, then we could figure out which field is contributing more energy or if they are contributing the same amount of energy. And turns out we can actually do that. We can express one in the form of other. For traveling electromagnetic waves, the magnitudes of magnetic field and electric field are related by this expression. That is B equals to E divided by C, where C is the speed of light. If this wave was moving in some medium, then it would not move with the speed of light. There would be a variable V over here. That is the speed with which it is moving in that medium. And we also know that C, this C is equal to 1 divided by the square root of epsilon not into mu not. If we think about the force that this magnetic field and this electric field would exert on some random charge on an electron, let's say, we will see that the force exerted by the magnetic field would be QVB, that is EV cross B. And the force due to the electric field would be the charge on the electron multiplied by the electric field strength. And if we compare these two forces, we can realize that the force due to the magnetic field will be less than the force due to the electric field, exactly by a factor of V by C. That is because if instead of B, if instead of B, we place E divided by C, this will just come out to be equal to charge on the electron multiplied by the electric field strength into V by C. So it is less than the force due to the electric field, exactly by this factor. And if the force exerted by the magnetic field is less than the force exerted by the electric field, it might be true that the energy that the magnetic field is contributing might be less than the energy that the electric field contributes. To check that, let's place this B over here. So we will place this B right over here and let's see what happens. So this now is equal to epsilon naught E square by 2 plus E divided by C whole square divided by 2 mu naught. When we work this out further, this will be total energy density equals epsilon naught E square by 2 plus E square divided by 2 mu naught into C square. And we know that C square would be 1 divided by epsilon naught into mu naught. So now we will be placing this value of C right over here. And when we do that, the total energy density, this is equal to epsilon naught E square by 2 plus E square divided by 2 mu naught and epsilon naught and mu naught will go to the numerator because C square is in the denominator over here. So you will have a mu naught and an epsilon naught at the top and one mu naught just gets cancelled right away. So the total energy density, this is equal to epsilon naught E square by 2 plus epsilon naught E square by 2. We see that the energy density of the magnetic field actually comes out to be the same as the energy density of the electric field. What we see now is so beautifully symmetric in electromagnetic waves. One is contributing the same amount of energy as the other. And we know that one cannot even exist without the other. The total energy density therefore comes out to be equal to epsilon naught. This is epsilon naught E square. And we can express this in terms of B if you would like that. Just simply by replacing in place of E you can replace C B and you can express the total energy density all in terms of magnetic field as well. And the units of energy density would be joules per meter cube. All right, so in this video we derived an expression for the total energy density in an electromagnetic wave by considering the energy densities of electric and magnetic fields. And we arrived at those by starting off from something that we have seen a lot, capacitors and inductors. And finally we arrived at an interesting point where we saw that the energy densities due to the electric and magnetic fields are exactly the same.