All right. So, here is an example from a movie Total Recall, which has a gravity elevator. It was released in 2012. So, it's connecting United Kingdom to Australia, UK to Australia through the center of the Earth. It's a scientific fiction. However, as you can see, when the elevator reaches the center of the earth, you see they're in no gravitational field, which is based on the physics that we just learned. Now, we're going to think about the question of, "Is the field of a point of charge exactly one over R squared? As an experimentalist, you may wonder if there's any error with the equation. Then, the next thing you have to think about is, whether if there is an error, what happens? If there's an error what happens? So, imagine a sphere here. Imagine you have uniform charge spread out on the surface. Then, you are thinking of arbitrary point inside a sphere, which is denoted as P here. Then, you can understand from the logic, that the part of this surface here, which is denoted as Delta A1, and part of the surface on the opposite side Delta A2, will exert electric field under point. If we integrate them, that will be the resulting force. So, because of the fact that Coulomb force depends on the charge density, as well as, one over R squared dependence, and the area of this patch depends on R squared. If you think about it, if it is exactly R squared, you will have no field whatsoever, but if it is deviating from R squared, you will have remaining fields. In that way, you can experimentally prove what the error is with the R squared exponent. So, here's how physicist or scientist proved the theory. So, here is a metallic sphere connected with insulating pillar and you have electrometer, to measure the number of charge deposited onto this test sphere. So, you charge this sphere first with positive charges. Then, you contact this outside of spheres with your metal test sphere. Then, you measure how much charge you have transferred from the device, and you can quantitatively measure them, but if you put it inside your metallic sphere, you always have no positive charge deposited. Meaning, to the sensitivity or to the resolution of the measurement of Paretas, R squared is pretty accurate. Now, what is the number here? If we put Epsilon as the error in the exponent, is less than one part in a billion. It's impressive. Usually, in experiments, Melody, what is the acceptable error range normally? To say, less than five percent. I'm not sure. Five percent, right? Five percent, three percent, but here one part in a billion is less than 0.000, a lot of zeros trailing one percent. So, that's amazing. Then, you may ask because every phenomenon can have size effect. So, until which size does this law is valid? What would be your guess? So, maybe angstroms scale? That's a good guess. But as you can see here, in the next slide, it is even smaller than angstrom. It is close to the size of electrons. Okay. So, we will see that. Let's see. So, in this slide, you will see Coulomb's Law is still valid, at least to some extent, as distances of the order of 10 to minus 13 centimeter. That's nuclear distance. For your reference, one angstrom, as Melody mentioned, is 10tilda minus 10 meter, which is 10 to the minus eight centimeter. So, even with our angstrom, if you think of this, it is 10 million times smaller. So, that's amazing. Now, scientists know that this law fails at the distance of 10 to the minus 14 centimeter. Either the electron or proton, or both, is some kind of a smear, like distributed charge. When we have distributed charge, we know it's not one over R squared anymore, it's R. If you can play with the distribution, you can think the dependence can be between R and one over R squared. You can design that, right? Good. Now, what if the spherical conductor is not a perfect sphere? Because we are making these spheres, right? There might be some aberrations. There might be some roughness. Then, does it matter? No. No. It doesn't matter. There's no fill inside a closed conducting shell of any shape and we can prove it. With a perfect sphere, it is easier to calculate what the fields would be if Coulomb had been wrong, but if it is right, then it doesn't matter what type of geometry we have. So, now we're going to move on to a next topic, the fields of a conductor, and here, we're going to learn a very important concept that is electric shielding. One-way and two-way shieldings, we will learn. Before going there, let's take a look at this arbitrarily-shaped metallic piece here. Imagine that you have the positive charge onto this conductor. So, they are spread out under surface. Then, we want to know what is the electric field out of this conductor. Again, we know from our previous lecture, that electric field inside the metallic or conducting materials is zero. So, we know it's zero inside. So, we know also, that tangential component should be zero. Otherwise, they will move around the surface. There will be no static charge. So, we know from that argument that tangential field should be zero. Then, the only one remaining is the normal field that is pointing upward. With that knowledge, we can create any kind of Gaussian surface, where you have a circular shape here, and the only phase that matters is the one that is outside the conductor. With that argument, you will understand that the electric field will be Sigma over Epsilon naught, which is the same field, the field between two planes sheets with opposite charge. It is two times that field of a one plain sheet. One way to understand that, is the charge here deposited, will exert field in both direction. However, the remaining charge will conspire to cancel this out, which is inside and at the field outside, to make it twice. So, that's what we'll learn. Now, the question here, which is in red font is written, if there can be no charges in a conductor, how can it ever be charged? We already gave you some hints here, but can you tell us, Melody? There can be charges on the surface. Exactly. So, that's the way we can charge a conductor. Okay. So where are the charges? They reside at the surface of the conductor, where there are strong forces to keep them from leaving. There are not completely free. So, Melody. Yes. What do we call this function that is keeping the electrons from escaping from this surface of the metal? Also which Einstein has published as photoelectric effect? The work function. Great. That's the work function for example for platinum, it is about 5.5 electron volt, depending on how you make them, but anyway, so they are not completely free, the excess charge of any conductor is on the average within one or two atomic layers of the surface which is commensurate with the Thomas Fermi screening lengths order of eight angstrom or few angstroms, right? And for our present purposes, it is accurate enough to say that if any charges put on or in a conductor, it all accumulates on the surface. So there are two-dimensional in nature. There is no charge in the interior of a conductor. Okay. If we use semiconductor is no more true, we will have depletion layer, we have accumulation layer which is no more 2D, it has 3D nature. All right. So let's take a closer look at the electric field outside a conductor, again, the same arbitrarily shaped conductor, and this is the blow-up image, which is on the right side, where we have cylindrical shape of Gaussian surface. As I told you, only electric field that matters is the one that is normal to the surface and out it is outside of the surface. we have local surface charges of sigma, right? Then, as I told you because outer surface has nonzero electric flux, all the flux will come from this one. And as a result, the electric field is sigma over epsilon naught which has no charge dependence as well. Then you may ask, why does a sheet of charge in a conductor produces a different feel than just a sheet of charge? Again, why does a sheet of charge on a conductor produce a different field and just a sheet of charge? And the answer is because the other charges surrounding the metal surface will conspire to make sure the electric field inside metals is zero. And make that field to add outside the metal surface. Okay? All right. So let's think about the fill in a cavity of a conductor. So, imagine we have carved out the inside of this arbitrarily shaped metal piece which was a deposited by plus charges. Then Melody, do you expect to have electric field inside this cavity? I don't think so. Because this situation doesn't make very much sense just looking at the picture. Okay, so she just gave us her hunch. We got feelings about this, there should be no electric field and I'll tell you the truth, she's right. But now, we want to know why in more scientific way. Okay? So we're going to prove it. How do you prove it? We can prove it by assuming we have electric field, and prove if we have electric field, we will have to violate the laws that we just discussed. So imagine we have electric field inside. Like from left to right. Then because we know in order to have this, we need two displaced positive charge to the left side and negative charged to the right side. Because we know the net charge is always zero inside metals, if you want to make a shield, you have to displace charged to one side to the other, and this is the only way you can think of. If this is the case, you will have electric field inside, but then, you can make a loop around it, including the part inside the metal piece. Then think if the law makes sense. So, we know from electrostatics the circulation of electric field will always be zero around the loop, or closed loop, but here because inside this metal, we have no electric field, the circulation will be zero. Along the line where we have electric field directed from left to right, we will have nonzero circulation. So that violates the law that we just discussed. Therefore, it doesn't make sense. Therefore, we know from this argument that there will be no electric field inside the cavity of a metal part. So, this is the concept of one-way shielding. So think about you're sitting in a car, it is a metallic can, and you're inside, and the tires are insulator, so it's perfectly floating, and say, there's a lightning strike outside, so it deposit a lot of positive charge onto your car, right? You don't have to worry about it. Why? Because inside, will not be perturbed or disturbed by this outside charge because you're shielded. So this is why when you have lightning strikes, stay in your car if you're in your car, right? However, one thing you have to think about is, what if I'm making a lightning inside my car, right? Melody, what if you're holding a net charge of plus inside a metallic can, and you have your friends outside. Will the friends outside feel the field from discharge? Yeah, I think so. Yes. Yes. So, the reason why is because the Gaussian surface will include net charge. So, even though there's no field inside the metal here, it will be penetrating through this metal, and influence your friend. However, the charge your friend has, here plus, will not influence your situation because inside your Gaussian surface you don't have any plus charge. So this is one way shielding. One-way shielding. Imagine now, you are grounding your car. So you connect it to ground, there are some cars where they have tailpipes with chains, metallic chains scrubbing the growth, maybe you saw that. Right? In this case, the ground make sure that the metallic can has the equal potential. So, in case you have plus charge here, it will gather minus charge outside the surface to make sure there's no electric field outside your car. So in this case, you will have two-way shielding. So depending on whether you ground your metallic can or not, you can either have one way shielding or two-way shielding. Okay. So, we're going to cover some important message or important knowledge on electric shield. So we have shown that, if a cavity is completely enclosed by a conductor no static distribution of charge outside can ever produce any field inside. So this explains the principle of shielding electrical equipment by placing it in a metal can. For example, your smartphone. All right. It is encapsulated, it is enclosed by a metallic case. So whatever happens outside your smartphones, they will not influence the inner components. Of course, if there's something happening inside your smartphone, that will influence outside. Okay. No static distribution of charge assuming no net charge inside, inside a closed conductor can produce any field outside. All right. So assuming no net charge inside. What happens if there is a net charge inside? What happens? Then it can be felt outside. Exactly. What if we ground the conductor? Then there's two-way shielding. Exactly. That is static, but knowing varying fields, the fields on two sides of a closed conducting shield are completely independent. Now, you understand why it is safe to sit inside a car when there's a lightning hitting the tree outside. Okay. So that's it for the lecture this week. Is it this week? And I hope you enjoyed the lecture, and we'll see you again. Bye. Bye bye.