 I'd like to briefly review the concepts of energy and conservation of energy because these concepts are fundamental to physics. After all, physics is the study of energy, matter, space and time. So this is one of the four foundational things that physics actually considers at its heart. You would have learned in 1307, the first semester course, that energy can never be created nor destroyed. It can only change from one form to another. So for instance, there's energy associated with motion, which is known as kinetic energy. There is energy associated with the potential to do work, for instance, so energy stored in a gravitational field that can be released later as work done by the field, converted into kinetic energy, for instance. So that's potential energy, and that's associated with something called the conservative force. But then, of course, there are other forms of energy. Heat is a form of energy. Fundamentally, it involves the kinetic energy of atoms. We know that, we've known that since basically the 1800s, that heat is a phenomenon which is due to the motion of the building blocks of matter. There's friction. So friction is at its heart simply sound. It has to do with the motion of atoms in one surface, moving past atoms in another. This sets up vibrations and resonances, and this causes energy to be dissipated from motion into the movements of atoms in the material. So there are all kinds of energy, and if one had the ability to sum up all of the ways in which energy can be stored or used in a system, so let's say there are n ways to do that, and you had the ability to add up all of those things, then the total amount of energy in a system, a closed system where energy can never leave the system and it can never enter the system from outside will always be a constant value, and that constant value can never be changed in total. The individual kinds of energy in the system can be turned into one another, but you can never change the total amount of energy present in a system. So this is at the heart of physics. It is not something we take for granted, it is an observed feature of nature, and this is known as conservation of energy. Energy is a wonderful concept. Energy is a scalar. It has no direction, it has only magnitude. Energy can be used to solve problems which are otherwise very much complicated by the presence of vectors, such as forces, or in our case, electric forces and electric fields. Those two things are vectors. They have both direction and magnitude. If we can find a way to express electrical phenomena in terms of energy stored or released, we can find a much more clean and simple way to describe the changes in a system that will lead to things like changes in the state of motion of parts of a system, such as charges present and responding to electric fields. So we will use conservation of energy as a foundational principle to move forward in understanding the electrical phenomenon. Now let's consider a very simple system. There is no friction, there is no heat. There is only the ability to store energy using a conservative force field and have the field release that stored energy as, for instance, kinetic energy. And a great example of a system that you have studied before is one involving a gravitational field. So right here on Earth, of course, we experience, we take for granted, in fact, a gravitational field all the time. The Earth is a very large body. It's orbiting a star. The Earth is rotating at all times. It completes one rotation every 24 hours. The surface of the Earth is moving quite fast. And if you remember anything about centripetal force and the tendency to want to move in a straight line when there is no other force holding you in place, you know that if you whip a ball on the end of a string around your head in a circle and somebody cuts the string, the ball flies off in a straight line. There's no longer a force from the string, the string tension, holding the ball in place. Well, we are like that tennis ball on the surface of the Earth. And at first it seems like there's nothing holding us here, that if the Earth is actually spinning and making one revolution every 24 hours, that we should be flying off the surface of the Earth. And of course, we're not. And that's because the mass of the Earth attracts the mass in our bodies. And the mass in our bodies attracts the Earth. And that mutual attraction through the gravitational force, action to the distance, keeps us on the surface of the Earth. The gravitational force can be described by a corresponding gravitational field. And that field is what is known as a conservative force field. This means that it has an associated potential energy. And the way, of course, I like to think about this is an analogy. So let's imagine that here is the surface of the Earth. So we'll draw this just as the ground. Here I am standing on the surface of the Earth. Yay. Now I have some kind of object in my hand. Maybe a tennis ball, not a bad example. So let's say I have a tennis ball that I'm holding in my hand. Now what I can do is I can hold that tennis ball at a certain height above the surface of the Earth. So let's say that the distance from the tennis ball to the floor is some height h. And of course I know from experience that if I let that ball go, then I am no longer exerting a force on the ball. And the ball is no longer exerting a force on me. I am no longer holding that ball steady in the gravitational field. So its atoms are attracted to the atoms of the Earth through the gravitational force. And it will begin to fall. So objects released in Earth's gravitational field, in fact as far as we know in any gravitational field, will tend to mutually attract each other. And the result from our perspective will look like the ball is falling toward the Earth. What's happening is that when I hold the ball there, I am storing in that gravitational field a potential energy mgh. And I'll just write this as u-gravity. So this is potential energy. This is simply the energy associated with a relative position in a field. So what's that position? So the position in the field is given by h. That's the height above the ground. G is the acceleration due to gravity. And specifically let's say at sea level because this does change as you change the height above the Earth from where you start your fall. And then finally of course we have m, the mass of the ball. The mass of the ball is the thing that's acted upon by the mass of the Earth. G, the gravitational so-called constant at sea level, that's determined by the mass of the Earth. And h is our height above the surface of the Earth. The potential energy, in fact all energy, is relative. There is no such thing as an absolute zero point for a potential energy. There is no way to measure absolute zero energy. We have to make all measurements of energy relative to some reference point. And so for gravitational fields it's very convenient to pick a reference point like the ground. You can define the ground to be the location of zero potential energy. So let me go ahead and do that. I'm going to declare that when an object reaches the ground that it is at u-grave equals zero. I declare that. Now when I'm above the ground I have u-grave greater than zero. In other words if I hold the ball above the surface of the Earth at height h I have now used chemical energy, I ate breakfast this morning. That breakfast is converted into fundamentally energy for cellular processes. My cells are capable of doing a variety of things. So for instance I have a muscle cells that will respond to electric currents that cause them to relax or contract. It costs energy to make my muscles relax or contract but because I have that energy stored inside of me from eating food this morning or from processing fat stored in my body I am capable of moving my arms. And so I can convert chemical energy into mechanical energy by gripping a ball and holding it and lifting it up in the gravitational field as I'm depicted doing right here. So I've taken my chemical energy and I've given it to the ball by raising it into the gravitational field against the place where gravity would actually like to move it. Of course gravity would like to move things down toward the center of the Earth and I've put something further from the center of the Earth. That costs energy from me. So in the me ball Earth system including Earth's gravitational field no net energy has come or gone but I've converted it from chemical energy to mechanical energy through electrical contractions of my muscles and I have now raised a ball up into Earth's gravitational field and held it there keeping gravity from letting it drop. And by doing this I have stored my chemical energy in the ball working against the gravitational field and I've thus added gravitational potential energy to the ball. So I must have a net positive gravitational energy. Gravity would like that ball to move toward the center of the Earth. Now of course it's always possible for me to get out of shovel and dig a hole so I could dig a pit right here. Now I'm still a height h above the top of the pit but there is now an additional height that goes from the top of the pit to the bottom of the pit and we can call this d the depth of the pit. Do I have to shift my gravitational potential energy zero point? No. I don't have to do that. I could do it if I wanted to because now when I drop the ball especially if I drop it over the pit it's going to fall to the floor of the pit. But I could also simply say that look when I go below the gravitational potential energy equals zero point then I've simply reached locations where potential energy is negative. There's nothing wrong with that. You can have negative potential energy. It's completely a relative definition. As long as you're consistent once you choose your zero point for gravitational potential energy or really for any potential energy you're totally fine. You can go ahead and have negative numbers and it's not going to make the world upload. So that's just a brief review of the concepts of a potential energy associated with a conservative force and its force field. Let's think about what the field wants to do. So the field wants to move objects from locations in the field of high potential energy to low potential energy. Absent any external forces on the ball like my hand gripping it, gravity would like that ball to fall. So if the gravitational field FG points down and I have a mass here M and that mass is acted on by no other forces except gravity. Gravity of course will accelerate it downward at 9.8 meters per second squared if you're at sea level on the earth and it will bring it to some later point down here which is at a lower relative potential energy to where it started. So we can call this point U initial, we can call this point U final for the associated initial and final potential energies. And what we should find is that the difference of the final minus the initial potential energies is less than zero because this is small compared to this which is bigger. Now that's if the field is acting on the mass, in this case the gravitational field acting on a tennis ball for instance. This is a feature of the work, the action taken by the field if it's a conservative force field. It will tend to want to bring objects from locations of higher potential energy to those of lower potential energy and the difference in the potential energies between those two points tells us the change in energy that the object has experienced. Now that energy has to go someplace and of course if you know from experience if you let a ball drop in a gravitational field you take gravitational potential energy and you convert it into kinetic energy. So there's a dance that happens here. This is a good example of a conversion of one form of energy into another. The ball held high in the gravitational field initially held by your hand has stored potential energy in it and no kinetic energy because it is not moving. But when you release the ball the only thing acting on the ball now is the gravitational field and it wants to reduce the gravitational potential energy of the ball. That energy must go someplace due to the principle of conservation of energy that we observed in nature and the only place that it can go because there's no other bucket in which to put energy is into motion. And so you turn stillness stored energy in a location in a field into movement and some remaining potential energy depending on where you are in the field. So to put this concretely in an equation form the initial kinetic energy and the initial potential energy added together because of conservation of energy must be equal to the final potential energy wherever you choose to measure that and the final kinetic energy added together. This here is just an expression of conservation of energy. There's nothing fancy going on here. Now you might remember that kinetic energy is given by one half the mass times the velocity squared. So if you know the velocity at a given moment for a mass m you can calculate its kinetic energy. If you know the kinetic energy in the mass you can calculate the velocity. That's one of the beautiful things about energy. Energy is a scalar concept. Velocity at its heart is a vector concept. They can be related to one another however and this gives you a bit of a hint of power of the scalar concept of energy. It's a number but that number can tell you about a state of motion and motion at its heart is a vector idea. It is both magnitude and direction. So let's think about our ball in the gravitational field one more time before we move on to electric charges and the associated electric potential energy that comes along with the electric force which is observed to be a conservative force. In our situation of dropping the ball we start off in a place where there is no kinetic energy and only potential energy. So initially I have potential energy from my picture above MGH and no kinetic energy. And when the ball hits the ground just before it touches the ground and stops it reaches a height of zero and according to my definition of potential energy there is no potential energy at a height of zero and that works out very conveniently with this formula when H is zero so is the associated potential energy in the gravitational field. So when the ball hits just as it touches the ground it has a potential energy of zero and all of its energy must therefore be in kinetic energy one half MV final squared. So in this particular scenario you can figure out how fast the ball will be moving just before it hits the ground because all the energy at the initial point was potential and all the energy at the final point is kinetic and so this is a very simple exercise in knowing the height and the mass. You can calculate the velocity and G is a constant at sea level it's 9.8 meters per second squared. In fact the mass is dropped out of this equation. You can see here for instance that V final will be equal to the square root of 2GH and that's it. So this is for a gravitational field and by getting the energy concept down and defining potential energy for the conservative force field you suddenly open up a whole world of possibility of simply thinking of everything in terms of changes in energy and then going back to the kinetic concepts of motion and location and time and acceleration and being able to relate energy to those kinetic concepts so that you can solve difficult problems using the scalar concept of energy. It's a very powerful idea and it's foundational in our understanding of nature. Before we move on to concepts involving the actual electric field I want to review one more essential ingredient in the discussion of energy and conservation of energy and that is the concept of work. Now quite generically if you have an object think of a box sitting on a flat surface like a table so just put some lines here to indicate this is a surface and you apply a force to the box. So for instance let's say you push on it and that force is directed to the right so here's f vector. If you can overcome static friction and change the resistance being due to static friction into the resistance being due to kinetic friction you know that you can displace the box. You can move the box along the line of force. So if your force is the overwhelming force that's acting on the object you will cause a displacement that is in the same direction as your force and we can denote that displacement as a vector d. The product of the net force that you exert on the system and d is equal to the work in this case done by you on the box displacing the box a distance d. Work has the same units as energy because it is a form of energy and of course those units in the m, k, s meters, kilograms and seconds system are joules which are written as a capital J. Force times displacement equals work and you can remind yourself of all the relationships between these things by remembering that force is written in newtons which is a kilogram meter per second squared. Displacement is of course in meters and so you are taking newtons multiplying by meters a newton meter is a kilogram meter squared per second squared and this is defined as three lines means defined as a joule J. This is the exact same units that potential energy has that kinetic energy has joules newton meters they are equivalent to one another. Kilogram meter squared per second squared all the same thing. Now with that basic general review of work in mind let's think about the work done by a conservative force field like gravity. So again we have some object like a ball and it is suspended a height h above the ground where we will define gravitational potential energy to be zero so I will denote that as u and this ball is immersed in a uniform gravitational field so here is f vector gravity. Now the gravitational field wants to move the ball from regions of high potential energy to regions of low potential energy. So we know that uf minus ui is going to be a number that is less than zero if the only thing acting on the ball is the conservative force field. This will be a negative quantity. Now we also know that when the gravitational force field does work on the ball it is accelerating it it is taking it from high potential energy to low potential energy that is converting potential energy into kinetic energy and so the ball will have more kinetic energy later in time than it did earlier in time. And so the work done by the field which will typically simply be just be written as w will be equal to the change in kinetic energy. Now the change in kinetic energy is also k final minus k initial but because it will have more kinetic energy later in time than it did initially this will be a number greater than zero. So how can we relate all of these things in a way that takes the signs into account that are clearly present a drop in potential energy a corresponding increase in kinetic energy and the answer is conservation of energy will do this for us. So all we have to do is write the energy equation for the total energy at the initial time u i and k i and the total energy at the end u f and k f those have to be equal and constant in the closed system these sums. So now we can rearrange and we can try to get u f minus u i on one side and k f minus k i on the other and see if we can relate these things well in order to get u f minus u i on one side say get u i over here I have to correspondingly move k f over here so this equation changes to k i minus k f equals u f minus u i which we know is delta u the change in potential energy well we're almost there except that we have the reverse of what we want we want k f minus k i and we have k i minus k f well no problem this is algebra we can just divide both sides by a minus one and that will take care of swapping k f and k i on this side. Equivalently we can multiply both sides by minus one it's the same thing so let's go ahead and do that let's multiply both sides by minus one so we have minus one times the quantity k i minus k f equals negative one times delta u the change in potential energy well if I distribute the minus signs I will get negative k i plus k f which is just k f minus k i and that's equal to negative delta u and remember the work done by the field w is equal to k f minus k i and so we arrive at a very important relationship for everything we're going to need going forward and I'll write this explicitly one more time the work done by the field which I will simply henceforth write as w is equal to the change in kinetic energy which is k final minus k initial and from conservation of energy we know that this is equal to negative delta u which is the negative of u final minus u initial so we have here a very important set of relationships let me highlight these there's a whole lot of stuff going on in here that's been equated one to another so let's just isolate a few key things here the work done by the field is the negative of the change in potential energy we know that potential energy when the gravitational force for instance a conservative force we know that potential energy decreases when the gravitational force increases the kinetic energy of an object by pushing it to a lower potential energy so it must be that the work done by the field which is positive the displacement of the ball is in the same direction as the force vector of the gravitational field f dot d is a positive number and so we must correspondingly have that the work is related to the negative of a negative number to get a positive number we have also that the work is equal to just the change in kinetic energy which if the only force acting on the ball is a conservative force due to a force field that's going to increase the kinetic energy so work will be positive and the change in kinetic energy will be positive so these will both have positive signs on on the left and on the right so these are a few key relationships that you need to keep in mind when we talk about these same energy quantities apply to the conservative force field that is the electric field now let's do some basic exercises with the concepts that we've just reviewed using the electric field and let me begin by drawing a uniform electric field pointing from left to right which I will denote e vector and remember we can write e vector as a magnitude e the strength of the electric field times i hat and here again e is a pure magnitude so it is a number that is greater than zero we can then also drop into this picture a charge so let's drop into this picture a positive charge so the positive charge which is this green dot it has a charge q which is greater than zero we can now write down the force that this external electric field will exert on this charge q f acting on the positive charge will be equal to q e vector which is q e i hat now since q is greater than zero and e is greater than zero this vector points in the positive x direction so the force exerted by the electric field on this positive charge which are written as f vector with a subscript plus that will accelerate the charge the positive charge to the right in the direction that the electric field is pointing nothing new here this is stuff we've already seen with kinetics motion and electric fields and accelerations now what I'd like to add into this of course is that in this field we have a displacement we have a displacement of let's call it delta x and this displacement of course has a direction associated with it so this is a vector delta x vector now this displacement vector delta x vector can be written as the magnitude of the displacement which I will put absolute value signs around so that we know that this has to be a positive number multiplying a unit vector in the direction of the displacement which in this case is also just I hat and so we can calculate the work done by the field on the positive charge using the classic force times displacement definition of work it's going to be Q e I hat dotted into the magnitude of the displacement times I hat again this is a positive number this is a positive number this is a positive number and these are unit vectors pointing along the positive x direction so we have three numbers multiplying a dot product of two vectors we can rewrite this just using algebra move the numbers out in front like this leaving the dot product and of course I would hope by now you know that when you take a unit vector and you take the dot product of it with itself you just get the number one so the work done by the field on the positive charge is Q e delta x so when an electric field does work to move a positive charge the work is a positive number w plus is greater than zero what happens if we drop a negative charge into this picture so here I have this light blue dot with a minus sign next to it the electric field is going to exert a force on this negative charge as well what direction is that force going to point in well if you said to the left you're correct negative charges want to move against the direction of the electric field lines because these electric field lines originate on some positive charge we don't see in this picture the negative charges are going to want to move toward that positive charge attracted to it by the electric fields from those charges and so there will be a a force on the negative electric charge that points to the left and displaces it in a leftward direction so of course here there will also be a corresponding delta x vector with a minus sign below it in order to calculate the work done by the field on the negative charge we just have to do the same exercises again so now let's consider a negative charge again being acted on only by this electric field well the force exerted by the electric field on the negative charge is going to be equal to Q E I hat E is a positive number but now Q is a number that's less than zero and so the way that I can algebraically and mathematically express this so that the signs are correctly accounted for like we're doing accounting like we're literally accounting and bookkeeping of of important things in our ledger I will explicitly write this as negative times the magnitude of Q so the absolute value of Q E I hat and indeed we see that this force points to the left absolute value of Q is a positive number E is a positive number the minus sign is now explicit and flips the direction of I hat from pointing to the right to pointing to the left and so indeed we have here a force acting on the minus the negative charge that points to the left what about the displacement well we can see from the picture that this is just going to be the negative of the magnitude of X I hat after all the displacement goes from right to left that's in the negative I hat direction and it has a magnitude of absolute value of delta X so now what happens when I combine this into the work done by the field on the negative charge well I have to take F vector minus and dotted into delta X minus and I get negative magnitude of Q E I hat dotted into negative magnitude of delta X I hat so I have essentially I have a negative one times another negative one times Q let me be careful about this times the magnitude of Q E magnitude of delta X I hat dot I hat I hat dot I hat is one and minus one times minus one is also one and so I find very interesting and very interestingly that since the magnitude of Q is a positive number and E is a positive number and the magnitude of X is a positive number the work done on the negative electric charge by the field is also greater than zero so quite generally I can say that work done by a field acting alone that is the only force in the picture is the force exerted by the net electric field in that region of space that work is always positive no matter the charge of Q Q could be positive Q could be negative nonetheless because of the way the electric field works to push positive charges in the direction of the force field a negative charges against the direction of the force field I always find that the work done by the field alone absent any other forces is positive and so if the work done by the field is positive what does this mean about the change in kinetic energy well let's consider the picture again I have a uniform electric field E I have a positive charge and it is displaced by the field which is the only force acting on the charge to the right accelerated in the process right word we know from conservation of energy which is independent of the force field that we're talking about that there is a relationship between the work done by the field and the change in potential energy and it's always that it is the negative of the change in potential energy that is uf minus ui this quantity is still a quantity that's less than zero so in order to get a positive quantity which is the work done by the field one has to multiply delta u by a minus sign and that all falls out of the conservation of energy discussion earlier nothing changes from gravity to electric charge but what does change is the fact that in electric field discussions unlike in gravitational field discussions there are two kinds of charge in gravity there is only one there is mass and as far as we know it's always positive but in the electric force there are two kinds of charges positive charges and negative charges and they respond differently to the same electric field a positive charge as drawn here will follow the lines of the electric field but a negative charge as we explored above will move in the other direction and so the force felt by a corresponding negative charge from the same field will push it in the negative direction nonetheless as we saw if you do the hard work of calculating the work done by the field you still find that w field because of conservation of energy is still negative delta u for that for that negative charge and that's still going to be equal to negative u final minus u initial if you do the conservation of energy equation for this problem there is nothing different between these two in terms of energy electric fields just like gravitational fields electric fields want to move charges of either sign from high potential energy to low potential energy it's just that the locations in the field of high potential energy for a positive charge are different than those for a negative charge for a positive charge this is high potential energy this is low potential energy so going further along the electric field line in the direction the electric field points for a positive charge takes you from a higher potential energy situation to a lower potential energy situation whereas for the negative charge going against the direction of the electric field takes you from the high potential energy situation which is over here on the right to the lower potential energy situation, which is over here on the left. That's the only thing that's reversed for these charges. What is meant by regions of higher and lower potential energy? So if gravity worked the way that the electric field does, and if there were two kinds of mass charge, positive mass and negative mass, we would find that positive masses would fall in a gravitational field on the earth, and negative masses would rise. Because rising takes them from a place that's considered higher potential energy to a place of lower potential energy. We've never observed this happening, so as far as we know, there is no such thing as negative mass, but it doesn't mean it doesn't exist somewhere. It just means that no instrument has ever been designed that's sensitive enough to detect the very little negative mass that's apparently left in the cosmos if there's any left at all. Now one last concept that I'd like to introduce before we go to the next lecture period is this concept of electric potential difference. This is a useful concept in the same sense that electric field was a useful concept. Remember that electric field was equal to force per unit charge. And so if we were talking about, for instance, a positive point charge and its associated electric field lines, and then we wanted to know something about the strength of that field as we went further and further from the source of the field, the positive point charge in this case, we could imagine dropping a tiny little probe charge into the field, q test, whose magnitude is greater than 0. And of course, if you wrote down Coulomb's law for this, f vector equals k q q test over r squared r hat, it depends. This force, of course, depends on the size of your test charge, no matter how tiny you make it so that it doesn't disturb the electric field of the primary thing you're probing. So it was convenient to define the electric field, which was the force per unit test charge in this case. And that divides out q test from this equation, leaving you with an equation that's agnostic to the test charge magnitude that you're actually using to probe the field. It leaves you just with the lines of force, but not the lines of force as they would be directed on a specific kind of test charge. And this charge independent concept is a very useful concept. And we're going to repeat this kind of charge independent concept right now. Let's begin by writing down the work done by the field on a charge q test. So imagine we have a picture like this, for instance, where you have a test charge that's positive and you've put it into the electric field of a point charge here, which now I'll be explicit about and write this as q greater than 0. So this is a charge q with magnitude that's positive. And then we have q test, which is also positive. And you're not going to look at what the field does to the test charge. Well, we know that the field will do positive work on the test charge, meaning it will move the charge from a region of higher potential energy to a region of lower potential energy. In this case, this means that the positive test charge will move further away from the point charge that's causing the field in the first place. So it will be displaced and accelerated away from the positive charge that's emitting the electric field in this case. And we know that the work will be equal to the negative of the change in potential energy. Now, the problem here is that we're talking about work being done by a particular charge on a particular test charge. And so again, it's very good to not have your energy definitions and discussions be slave to the size of the charge that you're talking about moving around in the electric field. And so we define something that is analogous to potential energy difference, but is independent of the charge that's being accelerated by the field. And to get this, all we have to do is think about what is the work per unit Q test that's being done by the electric field of the positive charge? And that's just going to be equal to the negative of delta U divided by Q test. So the work per unit charge done by the field in this case is equal to the negative of the change in potential energy per unit charge. All I did was divide both sides by charge. No magic here. It's a simple algebraic maneuver, not rocket science. But it allows me now to think about what it means to define energy in a way that's independent of the charges that are being accelerated by an electric field. And as you'll see, this is a very useful concept. And it gets its own name. So the change in this new quantity we're going to introduce called electric potential difference, delta V, electric potential difference. This is defined as the change in potential energy per unit charge. So the change in potential energy per unit charge is equal to delta V. And if we wanted to be really super explicit about this, Uf minus Ui, all divided by Q, by definition going to be equal to Vf minus Vi, the final electric potential minus the initial electric potential. So again, this is final electric potential. And this is initial electric potential, electric potential. That's the concept that we're defining now. It's potential energy per unit charge. So electric potential is potential energy per unit charge. And this is a very useful concept, as you'll see going forward, because it lets you make definitions of the relative changes in energy in an electric field without necessarily knowing what charges are moving around in the field. Now, it has its own units. So the units of electric potential difference are volts, and they are joules per coulomb. Energy per unit charge, joules per coulomb. That's a volt. So one volt equals one joule per coulomb. That is one unit of work per one unit of charge is one volt. Now, you can see that we can very easily relate electric potential differences to things like changes in potential energy, depending on the charges involved. And where your most common encounter with electric potential differences is, of course, batteries. Batteries have strengths measured in volts. So for instance, a 9-volt battery is capable of delivering 9 joules per coulomb that it moves. So if it moves one coulomb, it can give that one coulomb of charge 9 joules of energy. That's all that means. And you can relate these things simply by starting from this equation, delta v equals delta u over q. And if you want to figure out what the change in potential energy is going to be, thanks to this potential difference, like 9 volts, you can just rearrange this equation. Very simple. And get this one, that the change in potential energy will be equal to the charge times the change in the electric potential. That's it. Not so bad. As we get into this material in the course, things start to get a little bit easier. And you'll notice there are no vectors involved in this so far whatsoever. We have a vector-free thing because we're talking about energy. We're talking about charge. And those are all scalar quantities. You just have to make sure you get your signs right, and you'll be good to go. Now, finally, one last concept I'll introduce here before we get into the details of electric potential differences for different kinds of electric fields is a unit of energy that is also convenient. And that is known as the electron volt. In fact, this is one of the most common units of energy that you see in chemistry. It's one of the most common units of energy you see in my field because we're dealing with atomic and subatomic particles. And the energies of these things tend to be quite a lot smaller than one joule. The joule is no longer a convenient unit of energy for something like an electron. And so it's much better to talk about an electron volt. So what is an electron volt? Well, imagine I have a 9-volt battery. So a 9-volt battery has two terminals that look something like this. And the electric potential difference experienced by any charge between the terminals is delta V, and that will be 9 volts. So imagine I take an electron and I move it between the terminals of the battery. Well, an electron has a charge of 1E, which is 1.6 times 10 to the minus 19 coulombs. And it's negative. OK? So the electron volt is the energy gained by a single electron moving through a 1-volt potential difference. So an electron volt is the energy gained by an electron moving through delta V equals 1 volt. Well, what is that energy? Well, we know that delta U is equal to Q delta V. And this is equal to, let's just deal in magnitudes right now. So this is just 1.6 times 10 to the minus 19 coulombs times 1 volt. So this is equal to 1.6 times 10 to the minus 19. Well, a coulomb times a volt is a coulomb times a joule per coulomb. So you get a joule. So this is 1.6 times 10 to the minus 19 joules, a very tiny number. And this is defined as 1 electron volt, 1E times 1V, where E is the elementary charge. So this is a very helpful number. For instance, typical atomic energies, for instance, the ground state energy of the hydrogen atom are measured in electron volts. Roughly 13.6 electron volts corresponds to the ground state of the hydrogen atom. Energy transitions between energy levels in atoms by electrons are measured in also few electron volts or those levels. So it's a very convenient unit of energy for atomic things. And since atoms are the building blocks of nature, this is a very convenient unit of energy to discuss nature itself at its most fundamental scales. So you'll often be asked to convert between joules and electron volts. And this is how you do it. 1 electron volt is 1.6 times 10 to the minus 19 joules. So if you can remember elementary charge, you can remember 1 electron volt in joules. It's a nice conversion. So what we'll do next is we'll begin to exercise these concepts and look at, for instance, electric potential changes in the fields of point charges. And then we'll think about what it takes to assemble point charges. How much energy does it take to assemble point charges in proximity to one another so that we might imagine that now by combining the concepts of electric field and potential energy, we can begin to construct devices that store energy in the form of electric fields that we can then use to release later and do mechanical work.