Gradient Drift and Curvature Drift in Plasma Physics

[00:00:22]Hello, dear students. Today we will try to  understand; what is called as the Gradient   Drift and the Curvature Drifts. So far  in our discussions about plasma physics,   we have been analysing various types  of electric and magnetic fields and   howa single charged particle with a mass m  behaves in these different types of fields.  So, we have seen how uniform and static  electric field will influence the charged   particle we are also seen how a uniform magnetic  field will influence a charged particle. We also   seen a perpendicular electric and magnetic  fields how they will influence the charged   particles trajectory. We have drawn these  trajectories and with this understanding we   have moved on to the idea of a varying magnetic  field in space, but a static magnetic field.  So, in this process we have seen that if the  magnetic field is changing or if its strength   is changing from one point to another point. We  have taken a magnetic field such that its magnetic   field strength is lower in a particular region  and it is stronger it is at a particular region.  So, we have seen that the main influence of the  magnetic field strength changing will come in the   terms of its radius of gyration of the particle.  Since, we have known that the radius of gyration   of a particle is defined as mv by q B. So,  if the magnetic field is more the particle   will gyrate in a smaller circle; circular  path with a smaller radius and vice versa.

[00:02:07]So, what we have seen so, far is that if the  magnetic field is in this configuration such   that the magnetic field lines are farther apart  towards the negative y axis let us say. So,   the axis that we have taken was such that. So,  the magnetic field is grade is having a gradient   along the y direction and the magnetic field is  pointing outwards out of this page or we can say   that the magnetic field is along the z direction. So, in this configuration we have seen that   the particle will have a larger gyration  radius here and the particle will gyrate   in a smaller circle here. So, apart from this  basic the basic idea was the particles radius   of gyration changes as it moves across the y  direction this leads to the formation or this   leads to the idea of the drift velocity. So, this the difference in the radius of   gyration as it moves along the positive y  direction gives away the what is called as   the drift velocity fine. So, in this process  we made a simple assumptions saying that,   if you define the length of inhomogeneity  in the magnetic field with capital L.  We have we have assumed that this length of  inhomogeneity must be much greater than the radius   of gyration of a single particle; that means, what  we are trying to do is we are trying to make sure   the particle while gyrating in one orbit will  not experience any inhomogeneous magnetic field.  So, having said that we have try to expand  the magnetic field in Taylor series expansion   B 0 is the magnetic field strength at the  centre of gyration or the centre of orbit   then if it is the case: then we have seen  the magnetic field can be expanded as this.  So, if the magnetic field is expanded we  neglect all the higher order terms, then we have   calculated the average force that the particle  experiences along the three directions. Now,   since the magnetic field is along the z direction  and the if the particle is having a velocity v x,   v y, v z the particle will not experience any  force due to the magnetic field along the z   direction. So, F z is always 0 and F x was  evaluated to be 0 and F y was derived to be   q v perpendicular r L by 2 dou B z by dou y. So, I remember having stopped at this point   in the last lecture; that means, we have derived  the average force that a particle experiences as   it travels across this the across this gradient  or across this non uniform magnetic field. Now,   here our objective is to find the drift  velocity let us say we call this as v F   is we know that the generalized force will give  away drift velocity as F cross B times B square.  So, here F can be anything. So, while discussing  the average or the generalized drift velocity,   we have also said that the role of this F is to  give a non zero velocity to the particles. So,   that it comes under the influence of the  magnetic field. So, if we know v F is this   we call the drift velocity that arises due to  the gradient in the magnetic field as v del B   is equals to 1 by q since all other components  of force are 0 F y cross B divided by B square.  So, this F y the force that is experienced  along y direction is obviously, along the y   direction that is y cap is the direction  or j cap and the magnetic field is along   the z direction. So, if you substitute  the value of F y into this equation;   we can write 1 by q minus plus q v  perpendicular r L by 2 dou B Z by dou y,   y cross times B Z z cap divided by B Z  square. So, here we have to determine the   cross product between y cap and z cap and B Z  being just the magnitude gets cancelled here.

[00:06:53]So, we can write the velocity due to the drift in  the magnetic field as minus plus v perpendicular r   L divided by 2 B Z into dou B Z by dou y along  x cap that, so that is the basic idea. So,   basic idea is you have a magnetic field in this  direction this is B Z and this is Z direction this   is the x direction and this is the y direction. So, the magnetic field gradient is in this   direction the magnetic field is in this direction.  Now, we are able to see that the velocity of the   particles is in this direction, this is the most  interesting thing; that means, the velocity of the   particle is perpendicular to B as well as del B. So, this is the most interesting thing;   that means, the particle is moving in a direction  which is perpendicular to both the magnetic field   and the gradient of the magnetic field, which is  allowing it to move. So, this is the velocity..  So, now as you see r L is defined to be m v  perpendicular by q B. And we also know that the   magnetic moment is W perpendicular by B. So that  means velocity due to the plus minus that we have   velocity is positive is in the positive x cap for  electrons and is in the negative x cap for ions.  So, what does it mean? It means that the  gradient,we have a very important point if   we look back if we consider a magnetic field which  is homogeneous and static in nature; that means,   B dou B by dou x is equals to 0. I am trying  to make a point based on the particles moment.  So, in this situation were which was the first  or second particular case that we have discussed,   we have seen that both the particles; were  executing a simple harmonic motion or a   circular motion where the x and y components of  the velocity were giving you circular motion.   We have seen thatboth the particles will gyrate  along the same direction, but they will gyrate in   opposite direction in the sense the gyrate,the  circular motion was in the opposite direction.  So, let us say this is for the ions and this  is for the electrons, the radius of gyration   proportionate to the mass. What we have seen is  that both the particles will move in the same   direction ofcourse, their radius of gyration  will be depend will be different or they will   more be proportionate to the amount of mass.  So that means, electron being lighter particle   will have a smaller radius of gyration  whereas, the ion being a heavier particle   will have a larger radius of gyration. But, one important aspect is that this   uniform static magnetic field was not  able to create any charge separation,   but what you see when the magnetic field becomes  non uniform. Or inhomogeneous what we see is the   drift velocity is of course, dependent on the  charge; that means, it is different for ion and   it is in the different direction for an electron.  That means, because we see a charge separation.  That means so here the it is the arrow that is  drawn here is for the positive x direction. So,   ions are moving in these directions with the  velocity, electrons are exactly moving in the   opposite direction. So, what does it mean  this leads towards the charge separation.   So obviously, electrons and ions will go  in opposite directions become; that means,   that there is an electric field that will  be set up and ions will try to move across   this electric field and electrons will  try to oppose this electric field.

[00:12:01]So, the main important point is that the  magnetic field being in homogenous allows   charge separation. So, this is a important point.  So, the magnetic field it being inhomogeneous   allows charge separation and this the velocity  that with which this particles will travel will is   generally called as the gradient drift velocity. So, the gradient drift velocity can be more   conveniently written as let us say V del B let us  say, there are a few more points. So, plus minus v   perpendicular r L divided by 2 B Z dou B Z by dou  y along x cap. Now, for the given configuration of   having the gradient along the y direction. So, let us say the gradient is along. So,   the del B is in this direction and B being  just one component is in this direction;   for this configuration the particle trajectories  will look something like this, this is for ion and   this is for electron. So, what did I say ions  So, here we made a mistake according to the   science it will be positive x direction for  the electrons and so this is the direction,   this should be electrons and this should  be ions. So, this is ions and electrons.  Now what is the most important conclusion  about the gradient drift velocity is that this   separation of charges leads to something called  as a very important aspect in magnetosphere. Or   magnetospheric physics is leads to something  called as the planetary ring current we will   see how I mean how exactly and what is the  importance of this planetary ring current.  So, this planetary ring current is a very  important aspect of space physics. I hope   we canmake some connection between the gradient  drift velocity and the planetary ring current. So,   this was something about the gradient  drift velocity generally what happens   the gradient drift velocity can  be more suitably or simplified in   a more general form as so v del B is m v  perpendicular square divided by 2 B Z or.

[00:15:29]So, v perpendicular v del B is m v perpendicular  square divided by 2 q times B cross del B divided   by B q. So, now, for the given direction of  del B and for the given direction of B that   we have taken this form will again reduce  to the same equation that we have derived   earlier. Or this can also be written  as W perpendicular, W perpendicular   is the perpendicular kinetic energy divided  by q times B cross del B divided by B cube.  So, this is simply implies that the velocity  is perpendicular to both B as well as del B for   the case of earth the velocity will be in an  equatorial plane along the longitudes. Now, so   one very important understanding about themagnetic  field vector B is that we know from the Maxwell   equations that del dot B is equals to 0. So, what does it mean the first thing that   comes to your mind is that the magnetic field  monopoles will not exist; that means, the magnetic   field should always; so, one very important  understanding from this is that the magnetic field   should always converge and diverge. So,  if you take a simple bar magnet let us say   if we take a simple bar magnet the magnetic  let us say this is north and this is south.  So, the you will always see that magnetic field  lines will always be like this for a bar magnet.   So, one thing that you notice is that the magnetic  field lines are converging and diverging at the   poles. So, what is the direction of magnetic  field for this let us say direction of magnetic   field is the direction along which unit north  pole will travel so; that means, the direction   of magnetic field is from north to south. So, for the case of earth the magnetic field   lines diverge from the south-pole which is  geographic; that means, they will diverge   from the north-pole and they will converge to  the south-pole. So, this is the direction of   the magnetic field now what you see from this  figure is that del dot B is 0 of course, the   magnetic field lines converge and diverge at the  poles; that means, the strength of the magnetic   field is at the is maximum at the poles and is  its minimum or it varies as a function of r.  So, if you simply take a dipole magnetic field,  the dipole magnetic field will look something   like this the radial component B r is mu naught  by 4 pi 2 m by r cube cos theta. And B theta is   mu naught by 4 pi 2 m by r cube sin theta. So,  what does it mean, it means that the magnetic   field depends on the value of r. So, for different  distances the magnetic field will be different.  So, it means that the magnetic field any natural  any realistic magnetic field that you find will   always have a gradient. it is not possible that  you will have a uniform magnetic field let us   say for example, it may be possible, but over a  very limited space. If the magnetic field lines   are extending from infinity to infinity somewhere  in between you may find magnetic field lines which   are parallel and the magnetic field being uniform. But in all the other cases the magnetic field   lines are always curved; and they are  curved let us say between the two points,   between one pole and between the other poles; that  means, the magnetic field lines are always curved.

[00:19:58]So, what does it mean? So, the magnetic field  B is always associated with a gradient and a   curvature of the magnetic field lines. So, now what we have seen is that the   magnetic field due to its gradient gives away  what is called as thegradient drift velocity   we will also see that if the magnetic field.  Let us say if the magnetic field is assumed   to be this this curved magnetic field has a  radius of curvature let us say is assumed to   be R c now,a particle it is a charged particle  will gyrate across this magnetic field line.  Now as it gyrates it also experiences  another force in addition tothe force   that is created by the magnetic field the  additional force can be called as the F CF,   what is F CF? F CF is the centrifugal force  which acts away from the centre of the curvature.  So, due to the additional force F CF we  can also find out a drift velocity which   is called as the curvature drift velocity.  That means, any realistic magnetic field   will always be inhomogeneous and its magnetic  field lines will always be curved. That means,   if you see for example, if the magnetic field is  stretched over a very long distance then it may   be possible that this magnetic field lines  are parallel over certain spatial distance,   but otherwise the magnetic field lines are curved. So, if the magnetic field lines are curved it   will give away what is called as the curvature  drift velocity. Now the particle is of course,   gyrating parallel I mean the particles velocity  parallel to the curved magnetic field lines is   taken as v parallel. Now we againuse v R is  now the curvature drift velocity is again   F cross B by B square. Now F the is the  centrifugal force, which can be written   as let us say F CF is m v parallel square  divided by R c,which will act across R C.  So, v R the centrifugal force can be 1 by q times  m v perpendicular square divided by R c into R c   cross B divided by B square; or we can write the  unit vector R c as R c bar divided by mod R c.

[00:23:32]So, in that case we can write the curvature  drift as 1 by q m v parallel square divided   by R c square B square times R c cross B. So,  this is the drift that the particle experiences   as it travels across the curved field line. So, now the drift velocity due to the gradient   in the magnetic field is v del is written as m v  perpendicular square divided by 2 q times B cross   del B divided by B cube. So, this is again so, any  realistic magnetic field should always have these   two velocities associated with it any path. That  means, any particle which comes across a realistic   magnetic field will always experience these  two magnetic field, these two drift velocities.  So, these two drift velocities let us let us say  what do you see you see that the drift velocity   due to curvature. So, v R is perpendicular to  R c of course, and B and v del is perpendicular   to B and del B. Let us say for example, we put  the example of the earth. So, let us say if the   earth is like this. So, this is the geographic  south-pole and this is the geographic north-pole   and the magnetic field lines will look something  like this a realistic magnetic field lines.  So, the geographic north-pole is actually the  geomagnetic south-pole and geographic south-pole   is actually the geomagnetic north-pole. So, the  magnetic field lines should look something like   this the direction of this magnetic field lines  should look something like this. Now if you want   to understand the formation of ring current let  us define the directions let us say. So, if the   curved magnetic field line is like this the radius  of curvature is of course in this direction.  Now the gradient of the magnetic field is in  this direction away from the equatorial plane   let us say and the original magnetic field is  in this direction this is the B and this is del   B and this is R c this is the R c cap. Now so, the  possible direction for the drift of the velocities   the curvature drift should be perpendicular to R c  and B. So, it should be along the x direction and   the gradient drift should be perpendicular to B  and del B. So, it should be along the x direction.  So; that means, that when the particles come let  us say when the particle the plasma particle come  ,due to this gradient drift and the curvature  drift this will lead into charge separation;   that means, electrons will go in the positive  direction of positive x direction and ions will   go in the negative x direction; that  means, electrons will go to if the;   now the earth is revolving from west to east.  So, sun is in this direction this is the sun.  So, electrons will go towards the dusk  and ions will go towards the dawn. So,   this eventually creates charge separation  and this charge separation will produce what   is called as the planetary ring current earth  nearly 6 to 8 re it will be very strong current   in the sense that its magnitude will be of  the order of several millions of amperes.  So, we will talk more about this ring current and  its and its effects in subsequent classes. But,   however; it is reasonable that you combined the  drift velocity due to the radius of curvature and   the drift velocity due to the gradient and you  call this as the total drift velocity v total.  So, the idea is it is not possible to have  drift velocity only due to the gradient because   the gradient in the magnetic field imposes  that there should always be a curvature. So,   any realistic magnetic field should always  have the curvature as well as the gradient. So,   that then we define the total drift velocity of  a particle which experiences a magnetic field;   that means, any magnetic field will throw the  gradient part as well as the curvature part.  So, we can writethis as it is I mean we  can combine this drift velocities as it   is which is m v parallel square divided  by R c square B square times R c cross B   plus m v perpendicular square divided by 2  q B cross del B divided by B cube or we can   take a particular geometry and combine them in  a more convenient or in a more suitable form.  So, this is the total drift velocity that a  magnetic field will offer a charged particle.   So, if this drift velocity depends on the mass and  depends on the charge and depends on the magnetic   field. So, there is another more important way to  add these two drift velocity components by taking   a cylindrical geometry into consideration,  let us see we can do it in the next class.