 Hi, I'm Zor. Welcome to Unizord education. I would like to continue talking about electrostatic fields and we will consider today the concept of potential potential of electrostatic field Obviously, it's related to work, to energy and forces obviously which exist in the electric field Sometimes I'm Changing from electrostatic to electric which in this context is exactly synonymous Now this lecture is part of the course called physics for teen It's presented on unizord.com also every lecture is on YouTube and You probably can find it over there, but I do suggest you to Which the lectures from the unizord.com website because it's a course Which means lectures are presented in certain logical sequence every lecture has Detailed notes. There are problems solved. There are exams, etc. So and the site is completely free by the way There are no ads no financial strings attached. So Physics for teens electric field potential electrostatic field potential Okay, let me start from something which we have already learned First of all, we have learned that there is a Coulomb's law which describes basically the force which exists when two charges one of them at point a and another at point b located at the distance r Then each charge Experiences the force from another charge. Well, it might be attracting force if Charges are Similarly charged like excess of electrons in both or deficiency of electrons for both. The first one is called positive Excess not excess of electrons is negative and deficiency is positive Now if they are of different Types for instance one is positive another is negative then it will be the attracting force, but the Magnitude of the force is basically defined by This particular expression Now if we will assign Signs positive or negative to our charges again positive for deficiency of electrons and negative for Excessive electrons then the sign of this product would actually signify the direction of the force in One direction which is attracting or another direction which is repelling and Obviously it all depends on system of co-ordinate which one is to the right and which one is to the left But this is just irrelevant right now What's important is the sign is changing if both are positive or both are negative. It's always Repelling if it's positive and negative or negative and positive that's attracting. Okay, so that's what we know and It's about point objects right now. We're talking about electrical Electrically charged point objects Now the second component of the whole picture was the component the concept actually of of the electric field around the electrically charged object in case of a point object, it's a radial field which means There is a force which exists on certain distance from the point object which is the source of the of this field and this force is Acting on any other charged object which is brought into Into the field of the main object Now the force again, it's described as this one and then we have introduced The concept of intensity of the field you see when we're talking about the field. We're talking about one single Main source of this electric field now this is in the Coulomb's law We have two four two charges present in case of a field We're talking about the field produced by one particular charge and the way how it acts on another charge Which is brought into this field depends obviously on the charge of the main source and the charge of the probe So in this case, let's say it at point a we have the main force the main source so the charge of the Object at point a is called main and the point B would be a probe now How can we characterize the field of a single main object? quantitatively Well, if we will just take a probe object, which has a specific charge which is By definition is plus one Coulomb Then we can actually measure the force which the main charge Exords on this probe charge as the main characteristic of the field So if we know the intensity of the field Which is the force exhorted on this particular probe charge at any point in space around the main object charged with Qa amount of electricity This is the intensity of the field that completely defines the field itself as a characteristic of the this particular object which which is charged with Q a So Irrelevantly of any other charge The field and field intensities the characteristic of one particular charge at one particular point in space At some distance from this main charge. That's done too Okay, so what's next is next is what if we would like to move Some kind of a probe object. Let's say B from one place in the field Which is produced by the charge main charge to another field Now that's a very important thing because this basically is Related to energy because it's work right work related to energy So we have to spend some energy to to bring certain charges together or move it Far from each other, etc. So energy and work are related. So we are talking about right now about work Which is needed to move the object from one place to another. Well, let's go back to what is the work Well, obviously the work is in a simple definition The product of the force by the distance this force is acting well, obviously, it's a good definition but only for those forces which are going along The the trajectory and trajectory is a straight line and the force is constant without any problems Whenever we are talking about a little bit more complicated things like curved Trajectory and changing force. This is not good obviously, but what is good is we have to really talk about infinitesimal increment of work which we have to spend if we would like to use the force F on an infinitesimally small distance DS and Again here is if The force is acting along the trajectory. What if it's not? Well, then we have to multiply it by cosine of the angle phi We're on angle phi. So let's say this is trajectory so at this particular point this is my DS and this is a vector now and what if the force acts For instance at this particular direction, this is my force again as a vector So there is an angle here. So the projection on to this direction Would be the cosine of this angle, right? So that's why the cosine exists and Talking about vectors now, this is much more conveniently Expressed as a Scaliark product of Two vectors. So if you remember from the vector algebra If you multiply the magnet is of one vector by magneted by of another vector and the cosine of angle between them You will get something which is basically the Scaliark product of two vectors. So this is the most kind of general Definition of the work. So let's talk about the work which we can really perform moving object inside a spherical or radial Field produced electric field produced by some charge. Okay, so that's what we're doing right now Okay, so We have Let's use the capital Q instead of a Q a as a main source of electricity now So it actually All the forces from this are Directed in some direction Okay Now it doesn't really matter whether it's a positive or negative whether The arrow is this way or opposite when it's a traction or repelling doesn't really matter So let's assume that we have certain Charge q here lowercase q and this is the radius R between them So we know the force So the force is equal to K times q times q divided by R square Okay now So this is the force First of all, let's consider That we are moving our charge only Along the radius where it is So let's say we're moving from R1 to R2 From R1 radius to R2 radius Relative to center of Electrical charge my question is what's the work? Well as you see Even if our trajectory is a straight line our force is variable because the force is decreasing The force is decreasing as the radius is increasing, right? So we can't really just use f times s we have to really use some kind of a differential equations here or some kind of a calculus basically Which would involve integration now? What is this? Well, let's consider you have a point at radius x So let's consider x plus Gx so this is my infinitesimal distance My force we can assume that this isn't since Dx is infinitesimal increment. So the force depends only on the x. It's not changing within this infinitesimal interval now the distance is Between these main objects is K. So it's equal to q times q divided by x square now What do we do to calculate the work? We have to multiply f of x times dx which is the distance covered so f of x times dx this would be my differential of work and We have to integrate it to get the whole work. We have to integrate it from We have to integrate it from x equals to r1 to x equals to r2 f of x dx and this is the total work done by Done to Change the location from r1 to r2. Which is equal to of K q q x square dx Now this is a simple thing obviously integral of 1 over x square is Minus 1 over x because the derivative of 1 over x is minus 1 over x square So times this minus would be this plus. So this is actually K q q divided by x with a minus sign in limits from r1 to r2 Which is? R2 will be with a minus and r1 will be with a plus. So it will be K Q q times 1 over r1 So this is my w r1 r2 amount of work, which is needed to move my Charge to you lowercase q in the field produced by upper uppercase q from Distance r1 to distance r2 along the radius So this is a radial movement Great. Let me just write it down here. That's done So this is only for a radial movement Now let's talk about another type of movement Now we are in a radial Field right so what if we are moving the charge along the circle around or a sphere if you wish in a three dimensionless in a three dimensional space without changing the distance what happens if our probe charged with lowercase q is Moving along this particular trajectory, which is spherical around this particular Charge well, let's think about force Acts along the radius right so this is the force now my Trajectory is along the circle, which is this one Which is tangential to this particular circle now Tangent is perpendicular to the radius always Which means that this force is always perpendicular f is perpendicular to ds to an infinitesimal increment of Of the trajectory Which means there's color product is equal to zero, which means we don't really have to spend any energy There is no work, which is involved moving a probe charge around the source of electricity Because again the radius is always perpendicular to trajectory and there is no work But same thing is if you are in the gravitation field of the earth You are on the floor and you are moving horizontally something the gravity is completely neutralized and There is no work actually involved in this particular case, right Because the force is perpendicular now if you are in a Satellite which is circulating the earth On some orbit it just circulates by itself. There is no energy Which is necessary to move it along this particular Orbit well as long as the speed is properly calculated, etc. So it's not stanging still obviously So again to calculate the work Needed to move it from one place to another along a spherical or circular Trajectory around the main source of electricity requires no work now What's important so in this case we have the force times Times differential of trajectory is equal to zero. So the work is equal to zero now what's also important to To keep in mind that again in every field let's say it's a spherical field or regional field whatever Any kind of a movement Can be represented at any infinitesimal moment of time as movement along the circular orbit and movement regionally to another point now this is well, let's consider it on a On the surface not in a spherical point here again if I would like to move let's say from this To this I can move this way and then this way and this differential is Obviously easily represented as Two rep two differentials spherical and radial now What's important therefore is? Whenever we have any kind of a movement within this field We really have to understand that any increment can always be Implemented in spherical plus radial movement Which means considering that the spherical is always zero Right because we just talked about this forces perpendicular to trajectory and the radial is according to this formula We can always find out what exactly is the work Which is needed to move it from here to here it depends actually only on the radius This one and this one This is our one. This is our two And it's exactly the same as from here to here Because you can always do it this way and then this way now This is on a macro level. You can always do it on a micro level So for instance from here to here you can implement it this way a little bit here and a little bit here a Little bit there a little bit here again radial and Rageal a spherical and and and radial or circle and radial and that's how we will move Now what's important about this formula is Now if this is from R1 to R2 and then if you have from R2 to R3 Which is exactly similar to this if you will add them together What you will get? this will cancel out and WR1 R3 which is equal to this plus this would be equal to WR1 R2 plus WR2 R3 This is obvious from this right if you will add them together One two plus two three R2 will cancel and you will get exactly the same way as you would go as you would go straight from R1 to R3 So that means that the field is additive Well the formula for the work actually is additive and that And that actually results in a very important quality of the field No matter how you move from one point to another What's important is what's your beginning point and what's your end point actually the beginning? distance to the center and ending distance to the center no matter how Your trajectory is arranged because even if you go outside and then inside It would be positive in one case negative in another case whatever it is whether R1 is greater than R2 or R2 is Smaller than R3 or whatever Whatever else it will still be the same formula and the sum of all these movements would be exactly the same as Movement from the beginning to the end This is a very interesting characteristic of the electrostatic field The force within this field It's called stationary Stationary force and that's what's very very important. It's conservative the whole Field is called conservative field because it conserves the energy because this particular equation about the work means that No matter how you move if you will eventually come back to the initial point You will have the difference zero, which means your Work Will be will be zero no additional work is required. So this is a conservative Field in which any kind of a Movement in this field if there are no others Other forces are the fields involved Conserves the energy so whatever was before after you move within this field back to the original point Would actually not change the amount of energy The energy conservation is very important So the force is called conservative and the field is called conservative and And now we can go to a concept of Potential because now it's very important because what this looks like Well, it looks like the characteristic called V of r equals to k q divided by r is very important You can call this characteristic a Characteristic of the field and the potential of the field. So this is the definition of potential of the field That's basically the amount of work Which requires to bring your Movement you your your your object From the place where there is no energy, let's say it's infinitely remote place to the place which is at Distance r and the object obviously is plus one Probe object as we usually use the same thing as we have in Defined the intensity of the field. We use this probe object of one cologne positive one cologne to define the The Potential of the electric field. So it's a amount of work Needed to bring from let's say from infinity or any other point where there is no field Well infinity in case of a radial field obviously from infinity to To the Radius r now if you will put it this way Then the correct thing is our one is Equal to infinity our two is equal to r. Let's say and if you will Substitute it here one over infinity. Obviously. We are talking about limits here, etc. Will be zero So the whole thing would be was a minus sign So that's very important in the definition of the potential. There is this minus sign. So let me just correct I mean it's still important Without the sign, but the correct thing would be with a minus sign So the minus k times the charge of the central main object divided by radius is Called a potential of this electric field at the distance r from the center Which is obviously the same as any other on the same distance in the same radius now What actually follows from here is that if you move Certain Object from one radius to another radius What you really have to do is if you know the potential of the field You have to just stop subtract one potential from another because that's what this is if you're moving from R1 to R2 you have to subtract it from minus V of You have to subtract V of R2 minus V of R1 Which is what? minus K double K KQ capital Q Minus KQ over R2 minus again minus KQ Divided by R1 So that would be plus R1 like here and minus R2 and this is all for our probe object of unit charge for any other charge The difference between these should be just multiplied by lowercase q. So W R1 R2 Would be equal to V of R2 minus V of R1 times The charge which we are moving So knowing potential is sufficient to basically know everything about the field Because you understand what is the work required from to move one Object from one distance to another distance Depending on its charge, of course Now how about the force? Well the force is also very easy Remember how we have obtained the potential by integrating the force, right? Times distance. Well notice this if you have V of R Which is equal to minus K? Q divided by R What if you will Differentiate it by R. What do you have? Well the derivative of R is 1 over R square with a minus Minus 1 over R square with this minus it would be just plain R square, right? Which is intensity of the field So potential is Related to intensity by playing differentiating so knowing potential at every knowing this function V of R Potential at any distance R from the center It's sufficient to know the work which is performed to move a point from one point to another Put to move an object It's sufficient to know the amount The force the intensity The force actually is intensity times the lowercase q obviously but intensity of the field So you know everything about the field Okay position is given Intensity is given by this formula Potential is given by the potential formula this one So everything is interrelated So again Remember this interesting point about conservative force. So electrostatic forces are Conservative what else is conservative? Well gravitation same thing What's not conservative well can consider this what if you would like to move An object inside the water Well water resists the movement, right? Now that means that obviously amount of work to move if this is all Water to move from here to here is less than from here to Over there here. So even if gravitation Force produce basically a standard amount of But not produce it requires certain amount of work the Resistance of the water obviously requires different amount of work So the total work will be different depending on trajectory which are moving from a point to a point inside this resisting Substance so there are certain situations when The forces are not really conservative, but if you consider this to be in the vacuum electrostatic field, we are ignoring obviously the very small resistance of the air In this particular case if it's if it's on the earth in ideal situation in a model Which we are talking about a point object in in the in the center of the Radial field has certain charge then the field is conservative and the Conservation of energy is actually Fully in place in this case Well, that's all I wanted to talk about today Probably we will have to solve a few problems and I actually promise you to solve a few problems on potential on intensity of the field and That will be the next lectures. All right. Thank you very much and good luck