 Hello everyone, this is Dr. Rupali Sherke working as an associate professor at Department of Electronic Engineering in Wolchen Institute of Technology, Sholapur. In this video lecture we are going to discuss with the point form of the Ampere's Law and Strokes Theorem. Learning outcomes at the end of this video students are able to derive Ampere's Circuit Law which is also called as a Curse Law or Ampere's Law to the differential element and we are going to state with the Strokes Theorem. These are the contents which are going to covered in this video. Now let us recall what is the Ampere's Law. In the previous videos we have discussed with the Ampere's Circuit Law. Let us recall that yes what is the Ampere's Circuit Law? The Ampere's Circuit Law states that the line integral of the magnetic field intensity across the closed loop path is equal to the current enclosed by that path which is a mathematically given as a closed integral of h bar dL bar is equal to I. Now using this equation we will see the application of this law as a point form of the Ampere's Law. The point form of the Ampere's Law for that let us consider a closed path in a xy plane. We will consider a path following in a xy following in a xy plane and closing a current element which is along the z axis. The filament which is carrying the current will place that in along the z axis. Then the magnetic field is shown in a anti-clockwise direction according to the right hand thumb rule. So in figure we will consider a xy z plane in a Cartesian form and a plane lying in a xy plane. As it is lying in the xy plane we will consider the change in the length as a dx and dy change and we will consider a point P where we are desired to find the magnetic field intensity. Then the magnetic field intensity at a point P it is given by h bar is equal to h x0 a x bar plus h y 0 a y bar plus h z 0 a z bar. As it is in a Cartesian form it will be an x y z respectively. So according to the Ampere's Law it is a closed integral of h dL is equal to i. So we will expand this equation for this figure it is a closed integral will be now from 1 to 2, 2 to 3, 3 to 4 and 4 to 1. So the h closed integral h dL will be from h 1 to change in length l 1 to 2, h magnetic field in a 2 to 3 change in length l 2 to 3, magnetic field in a 3 to 4 change in the length from 3 to 4 and magnetic field from 4 to 1 and change in the length from 4 to 1. The magnetic field intensity closed integral will first find out the magnet this term that is a h 1 to 2 from 1 to 2. As we seen that this is in a z direction sorry it is in a y direction so it will be a h y 1 to 2 and change in length del l 1 to 2 will be now del y as shown in this figure because this is in a y direction so the length will be a del y. So we will elaborate this in a mode thing del y 1 to 2 del y will be equal to now this is the magnetic field intensity exactly at a point p this is at the point p it is a h y due to the this filament due to this filament this line it will be a as it is in a y direction that's why it is a h y 0 and the magnetic field is varying in this area in this space so it will be a change in the magnetic field with respect to x and for how much length it is a half of the del x so it will be a del x by 2. So the total magnetic field is the addition of at that point plus the change in the magnetic field for this space so we will substitute this value over here at this point and del y as it is so we will mark this as equation number 3. Similarly let us see for the 2 to 3 for 2 to 3 what is the variation it is in a x direction but it is in a negative x direction so from 2 to 3 the h 2 3 2 3 will be in a x direction h x 0 plus change in the magnetic field in this space in this space as it is with respect to x with respect to y it will be dou h x by dou y and for the for the length it will be half of the del x by 2 and in the direction it will be a minus del x similarly now we will see for the 3 to 4 3 to 4 again the direction is y but it is a minus y so we will first consider for the point it will be a h y 0 over here and for this space for this space again the change in the magnetic field for the with respect to x and the direction as we are consider it as a negative in a del x by 2 see the direction is in a negative x y del x by 2 and y is also a negative y now for the equation from 4 to 1 for the for 4 to 1 it will be again a change in the magnetic field with respect to x therefore it will be a h x 0 plus change in the magnetic field h x with respect to y minus del y by 2 and as it is in a positive direction from 4 to 1 that will be a del x so when we now as we are mark this equation as equation 3 4 5 and 6 we are going to substitute in this equation equation number 2 after substituting this in a equation 2 the equation 2 will reduce to see as as when we multiply this term inside the bracket this will be minus this will also be a minus the z h y 0 and h y 0 1 is positive and negative they will cancel out here h 0 and 1 h 0 is minus that is why it will also get cancel and only we remains with the 2 h y plus 2 h del x into del x by 2 into del y minus 2 del h x del y del y by 2 into del x when you then 2 to get cancel out over here and only you remain with the dow h x by dow x minus dow h z by h x by dow y and dy del x del y these are the common term from the equation close integral of h d l is equal to i according to the ampere's law so we will equate this equation received equation equal to i when you take this term at the denominator on the r h s side i upon del x by del x by del x is nothing but a area this is a area therefore which is covered in a x y plane this is a nothing but a current density a current density is defined as a ratio of current per unit area which is denoted by j as our current is in this we are assumed that it is in a z direction so the current current filament is in a z direction so the j will be the current will be in a z direction so we denote j z over here this is dow dow dy by dow x minus dow dx by dy if you see that our plane we are consider is a x y plane so the equation will be in a x y plane similarly if you consider the loop in a x z plane here we are considering in a x z plane then the equation will reduce in this will be in a equation will be in a x z plane x and z and the current carrying filament will be in a y direction similarly if you consider for the y z plane for y z plane the equation will be in a terms of y and z and the current density will be in a x direction so we got the equation seven eight nine we will combine so what will the j in a represented in a Cartesian coordinate it will be j x a x bar j y a y bar j z a z bar we as we know the equation j z j y and j x when we substitute in this equation number 11 so the equation will be this is nothing but a j bar will be equal to del cross h bar the curl product of the h bar which is a reduced equation which is nothing but a expanded form of the equation number 12 and this is a reduced form of this equation this is nothing but a called as a equation called as a point form of the Ampere's law which is you know given as a j bar is equal to del cross h bar where j bar is a current density and h bar is a magnetic field intensity using this we can state the Strokes theorem let us recall the Ampere's law which is a close integral of i d l is equal to i but i is nothing but a integral over the j d s because i is equal to j is equal to i upon s according to the Ampere's point law just now we are defined j bar is equal to del cross h bar when we substitute this equation number three in equation number two then equation two will be del cross h bar d s we are substituted value of the j bar comparing the equation one and two one and two on both the side it is compared with the i so it will be a close integral of h d l is equal to close integral integral over the surface del cross h bar d s so we can state that this is a Strokes theorem so it can be a any vector the statement for the Strokes theorem is given a integral of any vector across across the closed loop path is always equal to the integration of the curl of that vector this is an integration of the curl of that vector throughout the closed path so in general we can write this equation with considering the any vector a when you compare the divergence theorem and the Strokes theorem the divergence theorems relate surface with the volume and here it's relates the length with the surface and both the theorems are valid for the any vector field the curl can be represented in a Cartesian form that is nothing but a j bar del cross h bar is equal to by substituting the i is equal to one x y and z i j k and u v w multiplying factor as a one one one curl in a cylindrical form for that we will consider i is equal to rho j is equal to five and k is equal to z and u v w as a u is equal to one rho v is equal to rho and w is equal to one when this is a general form you can substitute the values over here and we can get the curl in a cylindrical form that is del cross h bar is equal to and in a spherical form for spherical coordinates we substitute r is i is equal to r j is equal to theta and k is equal to five and u is equal to one v is equal to r and w is equal to r sine theta in this equation and the equation will reduce in this form this is nothing but a j cross j bar is equal to del cross h bar these are the references. Thank you.