 Suppose we have a capacitance which is filled with air or vacuum and its capacitance is C0. Its distance between the plates is D. Now we are going to insert an insulator, a dielectric of dielectric constant K whose thickness is T. So it's not the same thing, the whole thing is not filled, it's partially filled. And our goal now is to figure out what's going to happen to the capacitance, what's going to be the new capacitance. So we need to figure out what the new capacitance, let's call that as CK. And we need to know what is this new capacitance in terms of this original capacitance. So it's going to be some number times C0, right? So that's what we need to figure out. So the question is how do we do this? How do we figure this out? Well we can always go back to our basics and ask ourselves how do we calculate capacitance in the first place? And one of the equations that does pop to my mind is remember the parallel plate capacitance formula which is C equals, I don't know if you remember this but epsilon not A divided by D. But I don't want to use this because this is a formula that only works for a capacitor which has no insulator in it. So I can't really use that for this, I'll have to sort of think about it, modify it maybe. So I don't want to use that, so I'm not going to use this because this is not the most basic formula for capacitance. I would like to go back to the most basic formula which is there's a definition of capacitance. Do you remember how did we define capacitance? Well we define capacitance in general, let me write that somewhere over here. Okay let me write it over here. Here okay. We define capacitance in general C to be the charge on one of its plates Q divided by the voltage across its plates. This was the general definition of capacitance. I like to use this because this will work regardless of what geometry your capacitor has, regardless of what medium you have put inside. This is the general definition. So this should work. So I'm going to start over here. So how do we use this? Well what I can do is first use this for our normal, our capacitor without the dielectric. Then use this for our capacitor with the dielectric and then compare and equate. Okay. So let's first use it for our dielectric, sorry our capacitor without the dielectric. Let me move it over here. So to use this I have to throw in some charge. So let me throw in some charge. So let's say I put some positive charge here. I put an equal negative charge over here. And let's say that charge is Q naught. Let's use naught for original values. Before I put the dielectric all the original values let's call it is naught. And let's say the voltage that was generated over here was V naught. So then I know that my capacitance, my original capacitance C naught is going to be Q naught divided by V naught. Okay. Next let's find CK, the new capacitance. For that let's insert our capacitor. Okay. Now what's going to be my new capacitance? Well that's going to be my new charge. I don't know whether the charge will stay the same or not. We have to figure out divided by the new voltage. Again I don't know whether the voltage is going to be same or not. We have to figure it out. Let's start with the charge. What do you think is going to happen to my charge? Will it stay the same when I inserted the dielectric or will it change? Now because we don't do this every day, it's not an everyday thing. It's not so easy for us to answer this question, right? One of the questions that we might even ask is how did we insert it? I mean in this particular, in this blackboard we inserted it like this. But how do we actually insert it? Doesn't that matter? Well the answer is it doesn't matter because remember capacitance, although it's like defined as Q by V, capacitance does not depend on the charge, does not depend on the voltage at all. It doesn't depend upon how you insert the dielectric at all. All it depends on is the geometry and what is the medium in between, right? The how part never matters. So we can decide how we want to insert it in a way that is like the easiest for us to calculate and then we can substitute. So the easiest way to insert our dielectric is just going to be carefully we just insert it literally like this. Imagine we literally inserted it like this. Imagine we are using our insulated gloves to ensure that no charge transfer happens to our hand. So if I use insulated gloves and I insert it like this, bam. And also let's imagine the capacitor is not connected to the circuit. Why? Because again, this does not depend on whether the capacitance does not depend on whether it's connected to a circuit or not. So to keep it simple, let's say capacitance, this is not connected to any circuit. Okay, under these conditions, what do you think will happen to the charge? Well, let's see. Because this is an insulator, this charge cannot move. So the charge has to stay. And my hands were also insulated. I was using insulated gloves. So the charge on this plate cannot move. The charge on this plate cannot move. That means the total charge stays the same. And therefore the charge on my new capacitor is still going to be Q naught. The big question is what's going to be my voltage? Will it remain the same or will it change? And I can guess that it will change. Otherwise, what's the point of the problem, right? If that did not change, it will be the same as the original value. Sometimes we get trick questions also, but I'm guessing it's going to change. All right. So how do I now figure out what the new voltage is going to be? Again, for that, I go back to my definition of voltage. Hey, what do we mean? How do you calculate voltage? How do you calculate potential difference? Remember, potential difference between the two plates is just the work done per charge. Let me write that down. Potential difference is the work done in moving a unit positive charge, or work done per charge, you can say. And we know how to calculate work done. That is force times the distance per charge. And force per charge is the electric field. And so let me use a different color for electric field. Let's use green. So the voltage or the potential difference is the electric field multiplied by the distance. And so now we have to bring electric field into the picture. So again, let's do the same thing. Let's first remove our dielectric and see what the electric field was. So let's get rid of our dielectric. Let's talk about what the original electric field was. So I don't know. Let's again put some numbers. So let's say there was an electric field like this. Electric field this way. Downwards everywhere. And let's say that electric field was E naught. This is E naught. And the original voltage or the original potential difference would have been the electric field E naught multiplied by D. So this would have been the original E naught times D. And now the question is when I inserted my dielectric, what happens to the electric field? Does it change or not? This is the point for you to pause the video and think about it. We have learned about this before. So pause, try and recollect and see if you can figure it out. All right. Let's recollect what happens to a dielectric or an insulator when you keep it inside an electric field. Now the charges can't move, but atoms can get polarized. What I mean by that is the positive charge of the atoms get repelled and the negative charge of the atoms get attracted to this. And as a result, they sort of start becoming this polarized tiny dipoles. They become this tiny temporary dipoles. And all the other atoms also tend to become like this. There will be another atom that does like this. And so that will be like this. Negative charge over here, positive charge over here. And as a result, notice we end up getting negative charge on one end and positive charge on the other end. And in between, they end up coinciding and sort of cancelling out. And so what does this do? Because of this, it creates its own electric field inside in the opposite direction. And that electric field tries to cancel the original electric field, the field which was in the vacuum. It doesn't quite cancel it. How much it cancels depends upon how much this is an insulator. But what this means for sure is that the electric field inside ends up becoming lower than what it was before because you're trying to cancel it. You're trying to oppose it so the total value becomes smaller. So the net result of all of this, we can get rid of all of these details. And to show that, I'm going to show electric field lines far away. That's how we represent lower fields, right? And if you are now wondering how much does the electric field inside becomes, how much lower it becomes, that's exactly what K is there to tell us. K represents how much it gets lower. It gets lowered by K. That is the meaning of dielectric constant. So the new electric field inside, the electric field inside becomes E0 divided by K. So the stronger the dielectric it is, the more it is able to cancel the field out and the lower the electric field inside. So this means that the total electric field has changed. The electric field has been modified. And as a result, the potential difference will change and so the capacitance will change. And so now the goal is to figure out what is the new potential difference between these two plates. If we do that, then we are done with the physics. Then we just have to compare these two or equate and figure this out. So now, how do I calculate what the new potential difference is? We can go back to the same thing. Potential difference is E times D. But because now we have the electric field here and here are different, what we can do is we can calculate work done in two parts. We can first calculate what is the potential difference between these two from here to here and then potential difference between from here to here and that total potential difference is just the sum of those two. So again, good idea to pause and see if you can try this on your own. Alright, let's do this. So the potential difference from here to here that's going to be the electric field here multiplied by the distance. The electric field I know is E0 divided by K times the distance is just the thickness of our dielectric. So that's the voltage from here to here plus the voltage from here to here. The voltage from here to here is the electric field here which is the same as earlier, the original field times this distance. What is this distance? Well, this total distance is D and this is T, this distance is D minus T. This is D minus T and I'm done. I'm actually, as far as physics is concerned, I am done. But I can't call this my final answer because this contains Q0 and E0 which were introduced by me. The original question did not have that. So the original question had C0. So now we now have to use our maths our mathematical abilities to somehow write this CK in terms of C0. So I have to somehow bring C0 over here. And again, that's also a skill. So again, great idea to pause and see if you can rearrange this and write it in terms of C0. All right. How do I do this? I see there's a Q0 over here. If I can somehow take E0 times D common then Q0 divided by E0 times D I can write as C0. So in fact, you know what I'll do? I'll pull out an E0. So let me tell you what I'm seeing. So this is Q0 divided by I'll pull out an E0 and I'll pull out a D as well. Because I know Q0 divided by E0D is just C0. And whatever remains inside just remains inside. Let's see what remains inside. So I had a T divided by K so that stays as it is because none of that got pulled out. T divided by K. Let me use blue for K. And I pulled out a D so there should be a D in the denominator. Plus I pulled out an E0 so that went. But I also pulled out a D so that remains. So I get D minus T divided by D. D minus T divided by D. And if you look at this carefully Q0 divided by E0D is C0. I can substitute that and I can get my grand and I think it looks ugly little bit ugly looking formula. I get C0 divided by T by K D oh gosh D minus T divided by D. And I know as ugly as it looks we are done. It doesn't matter how ugly it looks we are done. This is our new capacitance and by no means we should try and remember this because what matters is the thought process. Most of the times you will get a problem with some specific value which is given so all of these things will sort of like simplify but as long as you know how to do it then you will be able to solve any problem whatever value is given over here. And that's pretty much it. But before I go I want to leave you with a couple of questions to think about. One is what would have happened instead of inserting it at the corner what would if we had inserted this dielectric somewhere in between same dielectric constant same thickness would the value be the same or it would change. Think a little bit about it. Use the same logic and you will be able to arrive at the solution. That's one. The second question I have is this is not a question but this is a way to check whether our equation makes sense is think about a special case where T is equal to D meaning thickness is the same as the distance between the capacitor plates which means the whole thing is filled with the dielectric medium. What would this equation turn to? What would this equation simplify to and see if that equation makes sense.