in this video we're going to talk about capacitors so what exactly is a capacitor a capacitor stores electrical charge it's not the same as a battery a capacitor uses two metal plates separated by an insulator and it basically stores charge by taking electrons from one side and pumping it towards the other side the insulator could be air it could be paper it could be water anything that doesn't conduct electricity could be the insulator so that's basically what a capacitor is it's made of two metal plates separated by an insulator and it stores electrical charge now there are some equations that you need to be familiar with q is equal to cv q stands for the charge and the electric charge is measured in the unit's columns one column is equal to one amp times one second so q is equal to i t where i is the electric current in amps t is the time in seconds now c represents the capacitance and the capacitance is measured in ferrets v is the voltage measured in volts so what exactly is capacitance how can we describe it i like to think of capacitance in terms of charge efficiency one ferrat is equal to one coulomb per volt so let's say if we have two capacitors one has a capacitance of 10 farads and the other one has a capacitance of two farads let's call the first one capacitor a and the second one capacitor b now if we charge up capacitor a to a voltage of let's say one volt it can store ten coulombs of charge for capacitor b if we charge it up to one volt it can only store two coulombs of charge now what if we increase the voltage let's say if we charge up to 2 volts capacitor a can hold 20 coulombs of charge capacitor b if we charge it up to 2 volts if we connect it to a 2 volt battery it can hold up to 4 coulombs of charge so as you can see capacitance is basically charge efficiency it's how much charge you can hold per volt as you increase the voltage you can hold more charge but if you look at capacitor a it's more efficient it can hold ten coulombs of charge per one volt whereas capacitor b can only hold two coulombs of charge per volt so the higher capacitance means that you can store more charge per volt now q is equal to cv if you increase the voltage the charge will increase as you can see here however if you increase the voltage the capacitance doesn't increase the capacitance is based on the construction of the capacitor it doesn't depend on the voltage so make sure you understand that the capacitance is constant and it only depends on the construction of the capacitor now going back to the equation q equals cv let's talk about electric charge you need to understand that electric charge is associated with the quantity of charged particles and in the case of metals electrons are basically the charge carriers they're the ones that are free to move inside a metal the protons are fixed in place so the electric charge is equal to the number of electrons times the charge of each electron now the charge of each electron is negative 1.6 times 10 to the negative 19 coulombs so this charge is discrete that's the lowest charge that an electron can have every electron has that charge you can have a charge that's less than this number unless you have a fraction of an electron so charge is quantized now you need to know that the unit volt is one joule per column electric potential represented by v is basically the ratio between the electric potential energy and the charge q now be careful electric potential and voltage are not necessarily the same thing but they're similar voltage is the difference in the electric potentials of two points so voltage is delta v is the change in electric potential vb is the electric potential at position b v a is the electric potential at position a so electric potential is the electric potential energy per charge now electric potential and voltage they're both measured in volts so it's joules per coulomb which is one volt voltage is basically the ratio between the work and the charge it's the amount of work that can be done per unit charge so work is equal to q delta v and i believe there's a negative sign somewhere now let's get back to capacitance we said that the unit of capacitance is the ferret one ferret is a very very large number and only supercapacitors have this much capacitance most common capacitors that you may see like electrolytic capacitors they might be in the area of a microfarad it could be 10 100 microfarads you have some capacitors that are like nanofarads and even some in the picofat level a microfarad is one times ten to the minus six frats nano is ten to the minus nine pico is ten to minus twelve now there's another equation that you need to know c is equal to epsilon sub naught times a divided by d we said that the capacitance is basically a measure it's basically it's dependent on the construction of the capacitor so let's draw a capacitor here we have two metal plates separated by a distance d each plate has an area a and for a rectangle area is just the length times width the capacitance depends on the area if you increase the area the capacitance of the capacitor will increase because you can store more charge over a larger surface now if you increase the distance the capacitance will decrease given the same amount of charge if you increase the distance then the strength of the electric field between the two plates will decrease and therefore the electric force acted on the charges in between the plates if there is a charge will be weaker and also the capacitance will go down as well so make sure you understand this if you increase the area the capacitance will increase if you increase the distance the capacitance will decrease now sometimes you can add an insulator the insulator doesn't have to be air and if you add an insulator also known as the dielectric the equation will change c will be equal to k times epsilon times a over d now let's not forget this little zero here so k is the dielectric constant and for air k is about one point zero zero zero six is very small very close to one for a pure vacuum where there's nothing no gas molecules k is exactly one for other substances k will increase for example let's say if we have quartz for this material k is about 4.3 in the case of water k is about 80. now what effect does the dielectric have on the capacitance as you increase k the capacitance will increase so it's very useful to use a dielectric you can store more charge per volt c is equal to k times c sub naught where c sub naught is the original capacitance without a dielectric and c is the capacitance with the dielectric so anytime you add a dielectric the capacitance will go up however the voltage will go down v is equal to the original voltage divided by k so let's say if you have a capacitor that has 10 frads and let's say the voltage across it is 20 volts and the dielectric is one let's say there's air in between it now let's say if we add a material and the dielectric has a a constant of two i mean the insulator has a dielectric constant of 2. the capacitance will increase to 20 but the voltage will decrease to 10. so by increasing the dielectric you will increase the capacitance but you will decrease the voltage proportionally but notice that the total charge remains the same now you have to do this when the capacitor is charged but not connected to a battery because if you decrease the voltage of a capacitor and if it's connected to a battery then charge will flow from the battery to the capacitor bringing its voltage back up to 20. so you have to charge your capacitor first let's say if it's charged to a voltage of 20 then disconnect the battery and the capacitor and then add the dielectric when you add the dielectric when the capacitor is not connected to the battery the charge of the capacitor will remain the same the capacitance will increase with the new dielectric but the voltage will decrease so q equals cv notice that if we multiply 10 by 20 we're going to get a charge of 200 and that is 200 coulombs if we multiply 20 by 10 we will still get the same charge of 200 coulombs so by adding a dielectric to a charged capacitor when it's not connected to a battery the capacitance will increase and the voltage will decrease now i do want to mention something the equation that we had that i drew on a board that looked like this this is the capacitance of a capacitor if a vacuum is used as a dielectric if there's nothing in between the two metal plates now the equation changes to this if you have a dielectric it's going to be epsilon times a over d epsilon sub naught is the permittivity of free space it's 8.85 times 10 to the minus 12 column squared per newton per square meter so make sure you know that value because you're going to use it a lot now the other epsilon without the 0 is simply the permittivity of the material between the two parallel plates now epsilon is equal to k times epsilon sub naught so therefore we have this equation c is equal to k times epsilon sub naught times a over d that is if you replace epsilon with k epsilon sub naught so this equation is the capacitance of a capacitor if there's nothing in between the two metal plates if you have a dielectric then the capacitance can be calculated using that equation now let's talk about how to derive the formula for a capacitor so let's draw the picture of a capacitor here are the two metal plates one of the plates is going to have a positive charge and the other plate is going to be negatively charged and so there's going to be an electric field that flows from the positive plate and points towards the negative plate and these two plates are separated by a distance d now you can calculate the electric field if you know the voltage across the capacitor and if you know the distance between the two plates the electric field is simply the voltage divided by the separation distance now the electric field is also equal to the surface charge density sigma divided by epsilon sub naught and the surface charge density is basically the total charge on that plate divided by the area of the plate so starting with the equation q equals cv our goal is to solve for c so c is q divided by v and rearranging this equation if we multiply both sides by a we can see that q is equal to that does not look like a q q is equal to the surface charge density sigma times the area so let's replace q with sigma times a now using this equation if we solve for voltage voltage is equal to the electric field times the separation distance so it's e times the d now using the equation in the middle if we solve for sigma sigma the surface charge density is equal to electric field times the permittivity of free space so let's replace it with that so e times epsilon sub naught times a divided by e over times d is equal to the capacitance so now we can cancel c so therefore the capacitance depends on the area and the separation distance so that's how you can derive the equation now how does a capacitor work how does a battery charge a capacitor well let's draw a picture so let's draw the two metal plates of a capacitor and let's connect it to a battery this is a circuit diagram of a battery the long side of a battery is the positive terminal the short side is a negative terminal so this is negative and this side is positive now before you connect the battery if the capacitor is discharged it's going to have a voltage of zero and let's connect it to a 12 volt battery once you connect it there's a difference in potential whenever you have a difference in electric potential the voltage is not zero current is going to flow if the voltage is zero then no electric current will flow one way you can think about this is let's say if you have a level surface and if you have water on the surface this water will not flow the height between position a and position b is the same however let's say if you increase the angle let's say if you uh put on an incline and let's say a is at a higher position than b then water is going to flow from the high position to the low position and the more you increase the angle the greater the velocity it's going to what is going to flow down with more acceleration with more force the same is true with voltage if the voltage is zero between two points that is if the electric potential is zero between two points no current will flow current flows from a high electric potential to a low electric potential the same way as water flows from a high position to a low position so while the capacitor have a voltage of zero current is going to flow in the circuit now initially before we connect the battery at t equals zero the charge on the two plates is zero the number of electrons and protons are equal now let's say that on the first plate there's about a thousand electrons which means that there's a thousand protons for it to be neutral in reality it's probably much more than that you have like billions and billions of electrons and protons but let's keep it simple so we have a thousand electrons and a thousand protons and on the other side we also have a thousand electrons and a thousand protons now in a metal the protons they don't move but the electrons they're free to move so keep that in mind now once you attach the battery to the capacitor which the capacitor has a voltage of zero because there's a difference in electric potential current will flow now mind you current doesn't flow in between the two metal plates of the capacitor because you have an insulator there and insulators do not conduct electricity conductors conduct electricity so the electrons they will flow from one side to the other side now because this side has a positive charge the electrons on the left will flow in that direction and they will continue to flow on the other side so basically what the battery does using its voltage which you can think of as an electromotive force it basically pumps electrons from one side of the capacitor to the other side and that's how it charges it so over time let me draw a new picture let's say if 200 electrons travel from one plate to the other plate let's see what the situation will be so if the plate on the left loses 200 electrons it now has 800 it started with a thousand but it has 800 now the number of protons is still the same it's a thousand the plate on the right gained 200 electrons so now it has 1200 but the protons are still the same the protons don't move inside a metal now if you notice the left side has a positive charge because it has 200 more protons than electrons and the right side now has a negative charge it has 200 more electrons than protons so the magnitude of the charges on these two plates are equal but the sine is opposite so on the left side the charge is positive q there's 200 more protons than electrons on the right side the charge is negative q there's 200 more electrons than protons so the charges on these two plates will always be the same it's just that the magnitude is different now over time the capacitor will be charged to a voltage that's equal to the voltage of the battery so let's say at 12 volts it has a charge of 200. now 200 electrons doesn't correspond to 200 coulombs but let's just say the charge is 200 just to keep things simple how much more charge can we store if we double the voltage if we double the voltage then we can store 400 units of charge as opposed to 200 units if we triple the voltage we can store 600 the capacitance is basically the ratio between how much charge we can store and divided by the voltage level at that point so if you were to divide 12 by 200 it's going to be equal to 12 over 400 you're going to get the same value and that value actually i did it the other way around it's supposed to be charged over a voltage so if you divide 200 by 12 it's going to be equal to 400 over 24 and that ratio between the amount of charge that can be stored at a given voltage that's equal to the capacitance of the capacitor just keep in mind though this is not in clues that's just the difference in electrons and protons in reality to calculate c you need to find a charge in columns and then divided by voltage to do that it's going to be n times e so basically if you have 200 electrons multiplied by 1.6 times 10 to the negative 19 and then you can get the charge q each electron has discharged so just to review a battery charges a capacitor by pumping electrons from one side of the capacitor to the other side now once the is charged what's going to happen if we remove the battery and connect it to something that can absorb energy let's say a light bulb and let's say the capacitor has enough energy to light up the light bulb what's going to happen so let's redraw the picture that we have so this plate is positive and the other plate is negative now both plates still have about a thousand protons but the plate on the right has 1200 electrons and the one on the left is electron deficient as 800 electrons so even though the plate is positively charged in the left it doesn't mean that it doesn't contain electrons it simply means that there are more protons than electrons that's why it's positively charged and the plate on the right because it's negatively charged it doesn't mean that it doesn't have any protons it simply means that there's more electrons than protons so when you're dealing with electric charge think of it as the difference between the number of protons and electrons because all matter contains protons and electrons now electrons are going to flow from the negatively charged plate towards the positively charged plate and that's how a capacitor discharges itself that's how it uses up its energy electrons are naturally attracted to protons so if one side is negatively charged the excess electrons will flow towards the side that is electron deficient or that has a positive charge and as these electrons flow through the wires and through the light bulb the light bulb is going to light up provided that this capacitor has enough energy to get the job done and so that's how a capacitor discharges itself now the capacitor will stop working when the number of electrons are equal on both sides so once 200 electrons flow through the light bulb this will now be a thousand and this will increase by 200 so it's gonna be a thousand now at that point the two plates are neutralized they no longer have a charge they have equal numbers of protons and electrons so the charge on both plate is no longer positive q and negative q but rather qe is going to be equal to zero since the number of electrons are the same and the protons are the same so once it reaches the state of equilibrium the capacitor is basically dead it's discharged now there are some other equations that you need to know before we begin doing some problems and that is the electric potential energy stored in a capacitor there's three equations that you need to know the first one is one-half qv it's half of the charge times the voltage now we know that q is equal to cv so if we replace q with cv we can get another equation the electric potential energy stored in a capacitor is also equal to one-half cv squared so make sure you know these two equations one-half qv and one-half cv squared now there's also another one because we can replace v with q divided by c if you rearrange this equation so instead of replacing q let's replace v now so it's going to be one half times q times q over c so this equals q squared divided by two c so those are the three equations that you need to know to calculate the potential energy stored in a capacitor you