Biot Savart law (vector form)

we know current carrying wires produce magnetic field around them in this video let's figure out exactly how to calculate the strength of that magnetic field and it's given by a famous law in magnetism which is called the bo savar law that's how you pronounce them they're frenchmen so what does the bosavar law say for to use this law we have to consider a tiny element of the wire you can't just do the you can't just calculate it for the entire wire so let's say i consider a very tiny piece of that wire all right that very very tiny piece of that wire and let's say that that piece has a length of dl that's the length of that and i'm using d because it's very tiny you imagine it's an infinitesimal and let's say that the current is i and now imagine i want to calculate what the strength of that magnetic field is at some point over here some random point over here at distance r from this particular element so how do i calculate the strength of the magnetic field here so b or saw our law basically says that the strength of the magnetic field at this point and we'll call that d b b stands for magnetic field and d because it's a tiny magnetic field created by a tiny piece of wire so that's going to be vectorially so it's a vector equation get ready for this it's going to be a constant mu naught by 4 pi times i times d l cross r cap divided by r squared now i always found vector equations a little scary because you know they have all these cross products and everything so first thing we'll do is look at a magnitude of this equation and try to make sense of that so what would be the magnitude of this equation so if i just look at the magnitude of db which i'm just going to write db that's going to be this is a constant we'll get to that we'll get to what that mu naught is so mu naught by 4 pi i will remain the same denominator r square will remain as it is what is the magnitude of this how do you take magnitude of a cross b it will be magnitude of a times magnitude of b times sine of the angle between the two so magnitude of this that's just going to be dl times magnitude of this what is this by the way what is this r cap our cap is a unit vector in the direction of r and since it's a unit vector its magnitude is one so magnitude of this times magnitude of this which is 1 times the sine of the angle between them so sine of the angle between the dl vector how what is the direction of dl vector well you choose the direction of the current as the direction of the dl vector and you choose the direction of r as a direction of r cap so the angle between the two is going to be this if i call this theta then it will be sine theta because the cross product has a sine in it so sine theta so that's the magnitude all right so what is the equation saying well first of all there is a constant over here just like how we have constants in coulomb's law so there we had 1 over 4 pi epsilon naught here we have mu naught by 4 pi i'll get to that mu naught part in a second okay but there is a constant and then it says that magnetic field depends upon the strength of the current which makes sense because we know it's the moving charges that create magnetic field so higher current means more moving charges per second more magnetic field that makes sense we also see that the db depends upon dl the length of the current element why is that well that's because if you took a longer current element then you would have more moving charges in them and as then so the magnetic field will be due to more moving charges and so you would expect the magnetic field to increase so if you have double the dl you have double the number of moving charges and so you have double the magnetic field so this also makes perfect sense what does this say oh it's inversely proportional to r square that means that if you go farther away the magnetic field drops off as one over r square and that actually makes me feel very comfortable because we've seen this before in newton's law of gravity we've seen one over r squared in coulomb's law and now we're also seeing this in bosavar law so that's that's great and just like with coulomb's law how it only works for point charges we're seeing that bo sub r only works for point current elements so we have to only consider very tiny pieces of wire you can't consider that for the entire length of wire so so far you know everything was very similar to coulomb's law but there's one major difference there's a sine theta coming over here and that's a huge difference and let me show you how that difference you know actually pans out so if we go back to our charges and electric fields imagine i consider a circle around a point charge which is at the center and i asked you hey consider three points a b and c where do you think the electric field not magnetic electric field strength would be higher where do you think would it be the charge is at the center where would it be well because the distance is the same it doesn't matter everywhere electric field on this circle is going to be the same 1 by 4 pi epsilon not q by r square q is the same r is the same everywhere okay but now let's consider replace this charge with a current element okay a tiny piece of wire having current is called current element now i ask you again um where do you think would the magnetic field be higher would it be same everywhere would be different can you pause the video just look at this sine theta and see if you can figure out out of three places where it would be higher will be lower yeah there would be can you pause and try all right so this time we need to be a little bit more careful because we also have to consider the angle theta between the current and the r so the current is to the right what is the direction what what's where is r over here for this one our r is going to be from here to here okay here to here and so over here this is 90 degrees sine 90 is 1 that's the maximum value meaning over here you will get the maximum value of db magnetic field okay what about at this point well if i were to draw from here to here the r value remains the same but look at what happens to theta it's no longer 90 degrees it's more than 90. and sine of any angle more than 90 between 90 and some obtuse angle it's going to be less than one so db is going to be smaller what about over here well if i were to draw again uh r from here to here if i do that this time what's the angle the angle is zero what is sine zero oh sine zero is zero so over here our db is zero look at what what you're seeing even though we're going at the same distance the magnetic field is not exactly the same because it not only depends on the distance it also depends on the angle that's the most important thing over here we see that when you are at 90 degrees whether you are here or you are here somewhere 90 degrees you get maximum field and what we see is that at 0 or at 8 180 degrees you know if you go here this will be a 180 degrees sine wave is also zero you get zero and so in between you get magnetic field in between the maximum and the minimum value so it decreases and then it increases and then again decreases and then it increases again so magnetic fields will always be maximum perpendicular to the current element and they'll be minimum on the axis of that current element and that's what this this important thing is telling us all right now final couple of things one is what is this mu knot what's important is it's a constant for vacuum and most of the times we'll be dealing with vacuum and the value of that constant is going to be let me write that down over here the value of that constant is 4 pi times 10 to the power -7 and it'll have some units which you can work out this is tesla and you have current i don't remember the units i think you can work that out but it's given a name it's called per me a oops you can't i can't read this sorry per me a b t of vacuum and if you think hey that sounds very familiar to what we saw earlier yeah that was called permittivity this is called permeability i did not name it don't blame me i know these names are very very similar to each other what matters to me is that hey i know the value of that and it's it's 4 pi times 10 to the power minus 7. and so basically when you look at this whole constant the value of the constant is just 10 power minus n because 4 pi would just cancel out all right last thing let's look at the direction of the magnetic field because that's also important magnetic field is a vector it has a direction how do i figure out the direction of the magnetic field there are a couple of ways to do that i like both of them one is something that we've already seen before to find the direction of the magnetic field we can use our right hand clasp rule so you take your right hand and you clasp the conductor so that your thumb points in the direction of the current then the encircling fingers will give you the direction of the magnetic field and then you can use that to figure out what the magnetic field direction would be so everywhere to the right doesn't matter where you go everywhere to the right the magnetic field is into the screen so immediately i can say hey the magnetic field at point p should also be into the screen the magnetic field somewhere to the left would be out of the screen but we have a vector equation we should also be able to get the same answer just by looking at this vector equation so let's try that let's get rid of this class rule you can always use the class rule but you know using two two methods are always better you can always check yourself so over here uh if you want to get the direction of magnetic field you have to get the direction of dl cross r so here's how i like to do i have look at my dl it's this way i look at my r is this way so i have to do across from dl to r and how do you do cross from dl to r you take your right hand and you align it such that your four fingers are along this cross here's how i would do it so if i would show you my hand i would align it with my dl and then i cross it this way and while i do that look at the direction of my thumb the thumb gives you the direction of the cross product the thumb is pointing inwards and therefore the magnetic field over here must be inwards so both methods the clasp rule and the cross product will give you this same answer so can you quickly find out what will be the direction of the magnetic field at point a and point b can you pause and find that out all right if you use the clasp rule then we clasp our conductor such that the thumb points in the direction of the current that gives you the magnetic field encircling fingers gives the magnetic field i see that at point a the magnetic field is coming out of the screen so immediately i understand that the magnetic field over here must be out of the screen and we show out of the screen this way and over here everywhere down the magnetic field should be into the screen so everywhere below it should be into the screen but can we also confirm that using our cross product of course so if i start with say at point a i have to cross from dl to r so i have to cross this way so my encircling finger should go this way and so the way i align my palm is like this preparing it to cross and when i cross it with my circle my four fingers that's how it looks and so that look at the direction of the thumb it's coming out of the screen that's exactly what we predicted and similarly if i were to do at point b this time i have to cross it the other way around and so i'm going to hold my palm the other way around like this if i hold it like this now i cross it in this direction and so my thumb represents the magnetic field is into the screen it's exactly what we get over here and so now that we know how to calculate magnetic field due to tiny pieces of wire if you want to calculate the total magnetic field due to all the to the entire wire we just sum them up due to each tiny piece or we have to do an integral and we'll look at some problems in future videos