Free Physics NEET Video Lectures for Magnetic Effects of Current and Magnetism

00:00:00 e [Music] good morning to all of you in our earlier lectures we had discussed about electrostatics I had introduced the concept of charge we had seen that charge produces an electric field surrounding in the surrounding area that that electric field exerts the force on other charges so if you have similar charges they are repulsed if they have opposite charges they are attracted so we had seen the laws describing electrostatics we had also obtained gauss's law which helps us to obtain electric fields of charge

00:01:23 distributions and we are discussed in detail the fields produced by different charge distributions Now we move on to another topic which is magnetostatics so if you have a charge at rest then the electrostatic force on the charge is because of an electric field when the starts when this charge starts to move then we find that other than this electric field there is another force on the charge which is called the magnetic force so whenever a charge is in motion you can have an electrostatic force

00:02:01 because of an electric field and also a magnetic magnetic force because of another field which we defined as the magnetic field so this magnetic effects were discovered about 2,5 500 years ago when they found that certain pieces of metals attracted other metal pieces and so the field of magnetism was born and there have been a lot of experiments by various people to discover laws governing the laws of magnetism and their relationship to electricity so in this part of the module we will study things about magnetism magnetic

00:02:41 fields how magnetic fields are generated what are the forces because of magnetic fields and how we can employ them for various applications so before we start discussing magnetic effects I want to show you some demonstration simple demonstrations of magnetic effects which many of you may have seen somewhere in your in your uh during your studies okay so we will start with uh some magnets which I'm showing you here this is called a bar magnet and you can see here there is a on this on the upper side it's certain n

00:03:19 and on the other side is certain s n corresponds to North and S corresponds to South that's called a bar magnet there's another magnet here which is called the hor magnet n corresponds to North here ands corresponds to South here and you can have other shapes of magnets for example you have ring magnet here so you can have all kinds of magnetic magnets now for example I will show you that if I bring this what is written as n close to N here then the there is force of repulsion as you can see here if I bring

00:03:55 it to the point pole Mark s it gets attracted so just like Electric static forces which we had seen as having both repulsive and attractive forces this n attracts s but if I bring n close to N it seems to repel this so there is a repulsive force and an attractive force in these two cases you may have seen what we call as the magnetic compass here there is a magnetic compass here there is a magnet which is suspended on a on a rotating pulum and you can the the magnet can rotate and as you can see it

00:04:36 points along a certain direction irrespective of what I do to the mag to the uh to the system here when I rotate the needle always seems to point in One Direction and this is the direction of the magnetic field produced by Earth so Earth has its own magnetic field and this magnet gets aligned towards that magnetic field so when we study magnetostatics we will study what are the forces and torqus on magnets like this and how these magnets get aligned to different directions now in early stages of

00:05:11 development of electricity and magnetism electricity and magnetism were considered to be two separate Fields so electricity corresponded to charges and magnetism was described as Fields produced by magnets now it was in 1819 when H Christian oyer a Danish physicist who was giving a lecture and during the lecture he suddenly discovered that magnetic fields are produced by currents currents have a magnetic field around them so just to show you that experiment I'm taking a small mag uh battery and I'm taking a

00:05:46 wire so what I will do is I'll connect the wire to the battery and I will place this near the compass magnetic compass and I will show you that lead this leads to a deflection of the compass so let me hold the mag let let me hold the wire close to the magnet here and if you see the if you see the magnetic see immediately as soon as they connect the magnetic needle rotates showing that there is a force magnetic force on the coil on the uh on the magnetic needle and that rotates so this was an experiment which was performed by

00:06:22 H Christian oad to show that there is a very strong relationship between electricity and magnetism that currents produce magnetic fields so what is actually happening as we see as we will see is a current propagating through this as soon as I connect the battery to this wire there is a current flowing through the wire that current generates a magnetic field that magnetic field then affects the needle here magnetic needle and the Magnetic needle deflects so there's a torque on the magnetic needle which then deflects so this was

00:06:53 the experiment which was performed by Hans Christian oyed long time ago and after that experiment people like here Faraday Henry and all these people did a lot of experiments and the entire field of magnetism got developed now we have introduced the concept of electric fields at that time and I had introduced a unit we know that electric fields are measured in volts per meter so we need to have some unit of magnetic fields and as I will discuss when we start to look at the forces I will introduce a unit called Tesla this

00:07:30 Tesla is a unit of magnetic field and it is named after a scientist Nicola Tesla and it's a measure of the magnetic field Tesla is a very huge U unit so usually we use a smaller unit called gaus which is 10us 4 Tesla so I will introduce this later on again and I want to show you that there is a meter here which actually is called the Tesla meter which measures magnetic field of any um of at any point so I have now here two magnets which are extremely strong magnets as you can see here these are

00:08:09 very strong magnets which are which attract each other and you can see here very strong magnets here and I want to show you what is the kind of magnetic fields that these magnets produce so here is I put the magnet here and I this is a probe this probe at the tip of the probe there is a small Crystal which actually measures the magnetic field now here is the here is the magnetic field unit on that on this you can see here there is a let me zero the magnetic field here approximately 0.1 so Mt corresponds to

00:08:43 millitesla and that's almost 0o millitesla here so if I bring this close to the magnet as you can see here the magnetic field increases it's measuring the magnetic field and then there is a negative sign here which corresponds to certain orientation of the magnetic field if I take the Co sensor on the other side you see there's a positive value of magnetic field here and you can see the magnetic fields are quite strong here something like a 100 up to about 100 millitesla so this magnets are very

00:09:13 strong magnets and they produce typically hundreds of millit Tesla Earth has about 10 Micro Tesla magnetic field and these effects are very interesting from um and they they actually unify electricity and magnetism and incidentally there are naturally occurring living or natural living organisms which use magnetic fields for navigation for example there are bacteria called magnetotactic bacteria which have tiny magnetic crystals within them which help them to orient themselves along the direction of

00:09:49 magnetic field and this they use to navigate uh into the Earth because they want to go into regions with deficient oxygen contents similarly there are birds like pigeons which are understood to be using magnetic fields for the navigation for longdistance migration Birds use magnetic fields as one of the uh one of the sensing agents to direct themselves similarly there are ants which seem to use magnetic fields for navigation on the earth so magnetic fields are very very important aspects and we will in

00:10:22 this module study how the magnetic fields are generated by current carrying conductors what are the forces generated by magnetic fields and what are the applications of magnetic forces now I must mention here that we had in electrostatics we had introduced the concept of electric field as a vector field so we say that if you have a charge this charge produces a field surrounding itself which is called the electric field then if you place another charge here stationary charge then this electric field is exerting a force on

00:10:58 the stationary charge and either an attractive force or a repulsive force and that leads to the electrostatic force between the two charges similarly we will introduce the concept of magnetic fields so if you have a current carrying conductor if you have a conductor which is carrying current then this current carrying conductor generates the magnetic field surrounding itself in the surrounding medium that magnetic field then can affect a magnet like a magnetic needle here or another bar magnet or another current canning

00:11:26 conductor and it can apply forces which are called magnetic forces so just like electric fields we will introduce the concept of magnetic field which is also another Vector field and we will study various properties of magnetic fields okay so we just saw some demonstrations of uh magnetic effects so now I wish to discuss magnetic fields how magnetic fields are generated by current carrying conductors what are the forces that are exerted on other objects so so remember in electrostatics we had defined the

00:12:03 electrostatic force through an equation as follows so electrostatic we defined electric field as e is equal to F by Q so if you have a charge Q stationary charge Q then it is acted upon by a force F the force per unit charge we had defined as the electric field so this is the electric field now because there are isolated charges in electrostatics we could Define the electric field through an equation like this but we find that there are no such magnetic charges or there are no magnetic monopoles these are called No magnetic

00:12:54 charges or no magnetic monopoles so we will have to Define magnetic field through another relationship and that is and actually as I mentioned last uh just a while ago that magnetic forces appear only on moving charges so we need to Define magnetic field through another mechanism so suppose I take a charge which is moving in some direction then let me try to find find out what are the forces acting on this charge so suppose I have a region in this region I have a magnetic field for example a magnetic field produced by a

00:13:37 magnet or a magnetic field produced by a current carrying conductor and in that region I move a charge so let me assume that all objects around it are neutral so there are no electrostatic forces on this charge now we find that there is a force still acting on this charge because of its motion now what are the properties of these forces we find that as I change my direction of motion of the charge along one particular direction there is no magnetic force so if the charge moves for example in this direction suppose I have a

00:14:10 charge which is moving like this then there is no Force magnetic force but if it moves like this there is a particular force acting on the charge so there is one special Direction I find along which if I move the charge there is no Force there is no magnetic force if I move the charge along any other direction then there is a force acting on on the charge and that Force depends on so suppose this is the charge this is the direction in which F was 0 0 so this is velocity if I move like this it may happen that I will find there is

00:14:40 a force acting on this and that Force depends on this angle five between the direction in which I have zero force and the direction of motion and I also find that the force is perpendicular to the direction of velocity of the charge and to this direction of of zero Force so this Force which I see acting upon by on this moving charge is not only perpendicular to this velocity Vector of the charge but also the to the direction in which I had found that the force was zero so we Define a magnetic field B usually written as b along the

00:15:26 direction in which so the vector B is a vector which is oriented along the direction in which the charge found no Force so that's the direction of B Vector so this is the direction of B Vector along which the charge there was no force on the charge and we defined so we we then propagate suppose this is the direction I now propagate perpendicular to this direction this is perpendicular and I find I Define the force I find the force as f is equal to so I I I find certain force acting on this moving charge and

00:16:01 as I mentioned this force is perpendicular to the velocity vector and also the B vector and so I Define a magnetic field magnitude of the magnetic field as B magnitude which is given as B is equal to mod of the force let me write a v subscript tell me it's a magnetic force divided by Q * V so the force acting on this moving charge is why I so I I measure the force acting on this moving charge when the charge moves perpendicular to the direction in which I found the force had been zero so in

00:16:38 this direction I find the force and I the magnitude of the force divided by the charge which is moving multiplied by the velocity of this particle I Define as the magnetic force so in in terms of a vector vector field so I I this is the vector magnetic magnetic field FB Vector magnetic force is defined as q b cross B the magnetic force FB Vector magnetic force FB is equal to Q * V crossb Q is the charge of the uh the charge which is moving moving V is the velocity Vector of the charge and B is the corresponding

00:17:30 magnetic field so as you can see here if the velocity happens to be along the magnetic field V cross B becomes zero and the force becomes zero along other directions the force is finite so if the angle between the velocity vector and the magnetic field is five so if I have a magnetic field pointing like this and if I move like this this a velocity and this angle is five then the force magnitude of the force is equal to q v cross B magnitude which is equal to q v v sin 5 so at 5 is equal to 0 the force

00:18:18 becomes 0 at 5 is equal to 90° the force becomes maximum which is qvb and that is the way we have defined the magnetic field B so magnetic field magnetic force depends on the angle between the velocity vector and this magnetic field and varies like this qvb Sin Sin 5 so let me take an example so suppose I take a magnetic field so let me take a coordinate axis x y z so let me assume this this the direction of the magnetic field and let me assume the the charge is moving in the XY plane at an angle

00:19:00 five with the magnetic field so there is a plane I Define the XY plane as the plane in which which contains the velocity vector and the magnetic field so I can write the magnetic field v as B * J cap the velocity has two components it has a component along xaxis and a component along y- axis so I have V sin 5 I plus v cos 5 JC so the velocity Vector has is a vector so V sin 5 into x i cap plus v cos 5 J cap so the magnitude of the force magnetic force is q v cross B which is equal to q v sin 5 I plus v cos

00:20:00 5 J cross BJ which is equal to now q v b sin 5 into I cross J which is nothing but uh Q VB sin 5 K cap J cross J is zero so this component does not contribute to the force the only component that contributes to the force is the component along IAP so QV B sin 5 I cross JC which is K C so this Force as you can see here is acting perpendicular to both the velocity vector and the magnetic field and is oriented in a direction which is actually the cross product of B and B the magnitude of Force depends on the

00:20:59 angle five and of course the magnitude of the force also depends on the charge and the velocity also notice that depending on the sign of the charge the force is either positive or negative so if you have a positive charge the the force along K in this equation if the charge Q is positive the force is along K cap if the charge is negative it's along minus K cap now what is K cap K cap is this direction so one has use what is called as the right-handed rule for calculating the direction of the force on this

00:21:34 moving charge so the charge velocity is like this the magnetic field is like this so if I I use the right-handed screw right-handed rule so I take my I take my hand from I I take my right hand with the four fingers pointing along the velocity vector and move it towards the magnetic field Direction and the direction of the thumb tells me the force acting on a positive charge so I take a movement from the velocity Vector to the magnetic field the direction of the thumb tells me what is the direction

00:22:03 of the force now this is called the right-handed rule uh and this is also sometimes referred to as right-handed screw rule for example here I've taken a screw there is a nut here and there is a screw here so if I look here if I rotate in this direction if I rotate so let me take a screw like this if I rotate in this direction the screw is moving forward if I rotate in the reverse Direction the screw is moving backward so this this rotation in this direction gives me a force like this so if I take

00:22:36 the screw like this and I rotate the the screw from the velocity Vector to the magnetic field the direction of motion of the screw gives me the direction of the force so this is called the right-handed screw rule so I can either think in terms of right-handed screw rule or in terms of a right-handed rule so which means that I show my four fingers along the ve velocity Vector rotate the hand towards the magnetic field and the direction of the thumb shows me the direction of the force on a positive charge the force on the

00:23:04 negative charge will be exactly opposite so the force is perpendicular to the magnetic field this is very different from what you saw in electrostatics where the force was along the direction of the electric field so let me compare the two forces the electrostatic force and the magnetostatic forces so we can see here so let me take an example so let me take a charge of 1 micrum which is equal to 10us 6 k a magnetic field of 10 mesla millitesla and let me assume the charge is moving at a velocity of 10 m per

00:24:00 second so the force is equal to and let me assume that the velocity is perpendicular so this is the magnetic field and the charge is moving like this that's 90° so q v b which is equal to 10- 6 Kum into 10 m/s into 10 m Tesla which is equal to 10us 7 Newton that's the force acting on the charge so I have if I have 10 micrum which is moving in this direction here positive charge then the direction of the force as you can see is V cross B so I take my hand right hand from the direction of V to B and the thumb is

00:24:48 pointing downward so if the charge is positive the charge will be pushed downward uh because of the magnetic force so that's an example which tells me now I need to define the unit of this magnetic field so SI unit as I mentioned before it is Tesla this is after the scientist Nicola Tesla 18 1957 to 1943 so one Tesla is uh so I have to define the the force in terms of magnetic field the magnetic field in terms of force so 1 Newton by 1 Kum into 1 m/s which is also 1 newon by 1 Kum per second into 1

00:26:10 M and Kum per second is current so this is nothing but Newtons per ampere meter Kum per second is an ampere that is the unit of current so one Tesla is actually 1 Newton per ampere meter and that is the unit of magnetic field Tesla and as I mentioned to you Tesla is a very large unit so we also Define a much smaller unit called gaus so one gaus is equal to 10us 4 Tesla so that gives you uh the unit of magnetic field so let me give you some indication of the kind of magnetic fields found in various situations

00:27:08 so if you go to the surface of a neutron star the field is 100 million Tesla um in one of my early lectures I had mention mentioned about trains which are super fast called magnetic levitation trains in those trains we use magnetic fields of the order of five Tesla so these are trains which are floating because of magnetic forces and they can run at very high speeds magnetic resonance imaging is a very important instrument in medical field and this uses a strong magnetic field and the typical magnetic field is

00:28:06 about one Tesla near a small bar magnet which we saw a little while ago the the magnetic field is about 10 mesla the Earth's magnetic field is about 10us 5 Tesla and there is a magnetic field in the interstellar space and that magnetic field is about 10 Theus 10 Tesla so you see a very large large range of magnetic fields right from 10 Theus 10 Tesla in the interstellar space to near the surface of a star like a neutron star where the magnetic fields can raise to 100 million Teslas so it's a very large range of

00:29:12 magnetic fields and so you can generate very very strong magnetic fields using some of these concept that we will discuss so we now introduce the um a law which will tell me what is the magnetic field produced by a current carrying conductor this law is called bios law named after two scientists John Baptist B 1774 to 186 62 and Felix SAR 1791 to 1841 so they introduced this law which will help us to find out what is the magnetic field generated by a current carrying conductor now please remember we have

00:30:27 discussed in Statics that when you have a charge which is stationary it produces a field which we call the electrostatic field and that field then affects any other stationary charge so this is electrostatics because the charges are stationary now we do not have magnetic charges so we only have currents and so in a similar fashion we find that a steady current a current which is constant in time does not change with time will produce a magnetostatic field that is a magnetic field which will not

00:30:59 change with time now just like electric field e being a vector field which is a function of both position and time magnetic field is also a vector field which is a function of position and time and the magnetic field which is produced by a stitch with by a steady current that means I take a wire and I pass a constant current Through the Wire this constant current will generate a Time independent magnetic field in the surrounding space that magnetic field can then affect other magnets or other current can conductors or other charges

00:31:30 and as we have seen if there is a charge near a current K conductor and if the charge is not moving then there is no magnetic force on the charge even there even if there is a magnetic field because velocity is zero so we will introduce this bio law so we consider a let me consider a wire which is carrying current so let me consider a wire like this which is carrying current I so I want to find out what is the magnetic field at some point p so for this what I do is I take a small element of length

00:32:04 here DL DL Vector which is along the direction of the wire and let me draw the this this line to be along this dire this is the tangent to the wire at that point and let me join this so this is R vector and let me call this Theta so this current canning conductor means what there are charges flowing through the through the conductor and as we have seen a moving charge will produce a magnetic field so this moving charge in this which is current here produce the magnetic field so we will Define the magnetic field

00:32:41 produced at the point P due to the small current element as given by mu 0 by 4 Pi I DL cross R by R cu so the magnetic force magnetic field produced by a small current element DL Vector is given by I mu by 4i I DL cross R by R Cub R is the distance from here this is not a unit Vector so I if I want to write in terms of unit vectors I must write like this mu 0 by 4 Pi I DL cross R cap by r² so just like the electrostatic field this is also an inverse law 1 by R squ and it depends it's a vector it's a

00:33:32 vector field here and it depends on this quantity DL cross R so a small current element DL produces the magnetic field which is oriented along the vector direction of DL cross R and this quantity here this is introduced as a constant of proportionality so mu 0 by 4 Pi is a constant of proportionality and mu0 is called permeability of free space we had introduced epsilon0 in Statics pertivity of free space here we introduce another quantity called mu0 which is the per permeability of free space and this quantity has a value mu 0

00:34:37 by 4 Pi this mu by 4 Pi is defined to be 10 Theus 7 Tesla met per ampere so that's by definition the constant mu0 by 4 Pi is 10us 7 Tesla meter per ampere and mu0 is a constant proportionality so just like a stationary charge produces an electric field surround in the surrounding space a current carrying conductor produces a magnetic field and this small element of current DL carrying a current I so I is the current flowing through this wire and so this small element of current is I * DL Vector that current element produces a

00:35:17 magnetic field here DB Vector which is Mu by 4 Pi the current time DL cross R by RQ so just like a electrostatic fold field we have now defined the magnet static field in terms of current and in terms of the small current element now I want to make a comparison between the electric and magnetic fields as you can as you have seen here now so comparison let me put a comparison here both e and B fields are long range so they have they can act at very large distances the electric and magnetic fields they long-range

00:36:08 forces both of them decrease as 1 by R squ both of them satisfy an inverse Square law both obey princi of superp position which means that if you have two current elements producing two magnetic fields at a point then the total magnetic field in the presence of both current elements is the sum of the magnetic fields produced by each individual current Element e field is produced by a scalar charge which is the charge while B field is produced by a current element I DL a vector e is along the line

00:37:39 joining the charge and the point P while B is perpendicular to the plane containing R and ideal also the magnetic field B depends on angle between IDL and R Vector so these are some points which you may remember both electric and magnetic fields are long range Fields so they have effects over very long distances both decreases 1 by R square both of them are inverse Square law both the fields obey principle of superposition this is very useful to calculate magnetic field is produced by different current

00:39:00 distributions electric field is produced by a scalar quantity which is the charge while the magnetic field is produced by current element which is a vector IDL Vector the electric field is along the line joining the charge and the point P where you calculating the electric field while the magnetic field is perpendicular to the plane containing the r vector and the current element ideal and finally fin the magnetic field depends also on the angle between the current element IDL and R Vector now incidentally we may we may

00:39:37 note that epsilon0 mu 0 can be written as 4 Pi Epsilon 0 into mu 0 by 4 Pi 4 Pi epon 0 we have seen is 1 by approximately 1 by 9^ 9 and mu 0 by 4 Pi is 10- 7 so this is equal to 1x 9 into 10^ 16 which is equal to 1 by 3 10^ 8 squ and this is 3 10^ 8 m/s is nothing but velocity of light in free space so this is nothing but 1 by c² so it's important remember that c is actually equal to 1 by squ ro TK of epon 0 0 the Velocity in free space is related to the electric permittivity permittivity of free space

00:40:26 and permeability of free space to the this equation and that's very very important equation we'll come back to this later when we discuss Max's equations okay now I want to calculate magnetic fields of current distributions so let me take the following example I want to calculate magnetic field on the axis of a circular Loop of current so I have a I have a circular Loop which is carrying current okay so let me call these axis this is x- Axis let me call this xaxis this is y axis and this

00:41:31 is z axis so I Orient the axis at the center of the current Loop so that's a current Loop which is carrying a current and using bio law I want to calculate what is the magnetic field along the axis of this current Loop we will be able to find this through integration of the bio law for points off AIS it is not easy to calculate and we will restrict ourselves to calculating the magnetic field along the axis of this circular coil a circular Loop of current so how do we calculate so I take some point here where I want to

00:42:12 calculate let me call this point P so what I need to do is I need to calculate the magnetic field produced at this point by different current elements this element produces a magnetic field here this current element will produce a magnetic field here this current element will produce magnetic field here so I take all current elements in the circular Loop calculate the corresponding magnetic fields here and add them vectorially please remember magnetic field is a vector field so when I add the magnetic fields I must be careful in

00:42:43 acting them vectorially so I calculate the magnetic field from all the small small elements of currents and use the superposition principle to sum them up vectorially to calculate the total magnetic field now remember we had we had this bio law DB is equal to Mu by 4 Pi I DL cross R by RB so if I want to calculate the current due to this current element I draw a line like this this distance is r and this is IDL this small element is IDL this is the magnetic this is the r Vector this R Vector so remember this

00:43:29 magnetic field at this point because of this current element it is perpendicular to both the DL and the r vector and it is given by this equation now please notice here that DL and R are always perpendicular because of the orientation of because I'm choosing myself to be along the axis of this uh circular Loop so DL cross R magnitude is always equal to d l r r is the distance from here to here and the direction of magnetic field is perpendicular to both so let me draw another figure here to to show you the

00:44:14 orientation of magnetic field so let me take the uh x z plane so remember the loop the current Loop is along this Direction so the current is coming out from here I drop a point here and the current is going back into this that's the tip of the arrow that's the back of the arrow so the current is coming out of the paper here and current is going into the paper here so that's my point P where I'm to calculate the magnetic field so when I because of this element so this is R Vector DL Vector so DL cross R so DL

00:44:55 Vector is perpendicular to the page so the vector perpendicular to DL will be in the plane of the paper and that Vector has to be also perpendicular to R Vector so the magnetic field produced by this will be now I must use the right-handed screw rule so the current is going up coming out of the paper and I rotate towards R and I get the magnetic field direction as this this is the magnetic field produced by this current element at this point this is perpendicular to the current element which is coming out of the paper

00:45:27 because B Vector lies in this plane it's also perpendicular this V this R Vector Direction so if I call this Theta so B Vector now has both components along the x-axis and the Zed axis in the exact plane now it's interesting to note because the the problem is very symmetric so it's interesting to note that if I look at the current element which is exactly diametrically opposite on the other side so for example in this figure here if I'm looking at this current element I'll look at this current element if I look

00:46:00 at this current element there's another current element here for a current element here there's another current element on the other side so what I do is for a current element here this is the magnetic field for this current element which is now current is going inside the magnetic field happens to be exactly the same magnitude but in this direction because this current is going inside the paper and and the r Vector is here corresponding to this element the magnetic field happens to be in this

00:46:33 direction that angle is also Theta the current element DL and the distance are exactly equal for both cases so the magnitude of the magnetic fields at both cases are the same so let me call this s DB and DB so this is the DB current magnetic field produced by this small current element DL here IDL this is the magnetic field produced by another current element IDL here and what you know notice is they subtain the same angle with the Z axis and they are oriented like this so immediately I can see that this particular magnetic field

00:47:05 has a positive component along X this magnetic field has a negative component along X having the same magnitude but opposite directed so what you see is the X components of both the magnetic field produced by this current element and this current element will cancel each other and the Z components will add to each other so please just see this problem because of symmetry I find because I'm looking at the magnetic field along the axis of the circular Loop this current element produces a magnetic field oriented like this in

00:47:42 this direction oblique to the in the exact plane this current element produces the magnetic field in this direction the angles Theta are exactly the same because all the as you can see from the triangle here and because of this the X component of this magnetic field produced by this is exactly equal and opposite to the X component of magnetic field produced by this ele C element so what you'll find is if you take the circular Loop for every cent element there is exactly another Cent element in the

00:48:16 opposite diametrically opposite point which will produce another magnetic field whose x components will cancel so similarly here this component will cancel with this component the so all comp all the components of magnetic fields perpendicular to zed axis will cancel each other so this one and this one will produce magnetic fields whose components perpendicular Z axis will cancel similarly this one and this one will produce magnetic fields whose components perpendicular Z axis will cancel and so on so all that will happen

00:48:45 is all the components along the Z axis will add to each other and the components perpendicular Z axis will cancel from each other so all I need to do is first thing I notice is the total magnetic field produced by this CLE Loop must be along the Z axis and I can calculate the magnitude of the uh magnetic field now so let me write this as d b z is equal to so I written this equation here mu by 4 Pi so I have mu by 4 Pi i d l r by r R Cube was the DLR DL cross R is DL * R by R Cube and I must looking

00:49:34 I'm looking at the Z component which is cos Theta so this is the magnitude this is the total magnitude of the magnetic field and his it Z component is cos Theta this the X components cancel each other so that is the magnetic field here and cos Theta I can calculate so if this is the radius of the coil and this distance is R so cos Theta so this angle is Theta so notice that this line is perpendicular to this line and this line is perpendicular to this line so this angle is also Theta this R Vector is perpendicular to

00:50:10 this magnetic field is perpendicular to R vector and this line is perpendicular this line so that angle is Theta so cos Theta is nothing but capital r by small R so DBZ becomes mu I by 4 Pi r² d l into so I can write the magnetic field not i r by 4 pi r cub to D and if you notice so if this distance is Z then r² is equal to r² + z Square so I can write I can use this formula in this equation and write this as Mu i r by 4 Pi into r² + z s is^ 3x 2 DL so that is the magnetic field produced by a small current element

00:51:37 DL suppose this element now I must integrate over all the elements of the current which is along the circle so I will integrate this to get the total magnetic field DB bz is equal to Mu i r by 4 pi r² + z s is 3x 2 integral DL and integral DL is nothing but the circumference which is 2 pi r r by 4 Pi r² + z s 3x 2 into 2 pi r so this this gives me mu I r² by 2 * z s + r² so I can write the total magnetic field so if this is my current Loop this Z axis X Y so the total magnetic field along the axis at a point on the axis at

00:52:54 a distance Z from here is equal to Mu I r² by 2 * z s + r² 3x2 K okay so we can see we can actually calculate the magnetic field at any point along the axis and it depends on the distance from the uh from the plane of the circular Loop to this equation so if I to plot B magnitude versus Z what you'll find is uh as you can see here there is z sare + R square in the denominator the maximum magnetic field will appear when Z is equal to Zer and as Z increases to either positive or negative side the magnetic field will

00:53:49 decrease and so you'll get mag field going like this so this is the peak of the magnetic field which is given by so uh magnetic field at the center of the loop magnetic field B is equal to Mu i p so this is my current carrying conductor Loop here so at this point magnetic field is pointing like this so as you can see here again we have the right-handed screw rule here so if I if I take my if I take my nut like this so if I if I rotate if I rotate like this in the in the direction of the current I I see

00:54:54 that the screw is moving towards me and so that is a direction of the magnetic field so the right-handed screw again rule gives me the direction of the magnetic field so if I put my fingers along the direction of the current I get the direction of magnetic field so magnetic field is pointing along the K Direction here which is given by this now so this is for a single Loop if you have multiple Loops you can actually calculate so if you have n Loops closely Mound the total magnetic field will be

00:55:26 at the center will be mu not n i so you can actually increase the magnetic field by changing a large putting a larger number of Loops in the uh in the coil and you can get a strong magnetic field so let me take one uh calculate one example so let me take a a loop of radius 20 cm uh number of turns is 100 and the current I pass is 5 amp so the magnitude of magnetic field at the center is given by mu not n i by 2 R which is equal to 4 Pi 10us 7 into 100 into 5 / 2 * 2 which is approximately 1.57 mesla so you can see if you have a 100

00:56:40 Loop coil with 20 cm radius you will get about 1.6 mesla at the center of the coil and as you move away from the center on either side the magnetic field will decrease and notice also that the direction of magnetic field so if in the in this Loop here here the magnetic field is like this and the magnetic field is pointing away from the loop here so the current is Flowing like this now let me leave a small problem to you calculate the magnetic field at the center of a circular Arc of wire of radius r

00:58:03 carrying I so you have current carrying circular Arc that's a center and let me assume this angle is five so that's an arc so this is just the arc inside of a circle I have an arc carrying a current so what is the magnetic field here please calculate this thank you very much oh [Music]