Engineering Maxwell equations, EMwaves (MR).pptx

Engineering Physics

1. Module 3: Maxwell’s equations & E-M waves.

2. Scalar product If and are two vectors and magnitude of the vectors is and respectively, then is the dot product of and . In three dimensional representation scalar product of and is =  

3. Cross product Let there be two vectors and , the interaction of two vectors result in the production of vector and and is perpendicular to the plane containing the directions of and .  

4. Vector operation   is a mathematical operator called del (sometimes as nabla ) The operation is as per the equation are the base vectors. (Base vectors are same as unit vectors but oriented strictly along the coordinates in the coordinate system and pointing away from the origin).  

5. T can act. On a scalar function T called gradient. 2. On a vector function via the dot product . called the divergence. 3. On a vector function via the cross product x called the Curl.  

6. Gradient If there is a difference of potential between any two point charges in the region, an electric field exists between them. Let V be the potential set up by the charge and is a scalar quantity. The potential decreases as distance from the charge increases. Rate of change of potential decides the strength of the field . Relation is given as (- ve sign indicates the field is directed in the direction of decreasing potential) Hence (pronounced as grad V)  

7. Divergence Divergence of a vector field at a given point P is a measure of net outward flux per unit volume as the volume shrinks to zero about P and mathematically written as = Other wise written at P  

8. Physical significance of divergence The divergence of the vector field at a given point is a measure of how much the field diverges from that point.  

9. Curl Curl of a vector field at a given point P means, it is the maximum circulation of per unit area as the area shrinks to zero about P.   Curl of =     Also written as

10. Physical significance of curl The curl of a vector field at a point P is a measure of how much the field curls (circulates) around P. Let the field be magnetic field around a point P (Fig A, B and C). Fig A: the magnetic field vectors curl around P when a current is passed through a straight conductor. Fig B :Magnetic field vectors have larger magnitude all around P - a more rigorous circulation. Fig C: Field vectors have larger magnitude, but they do not have any turning motion : .  

11. Stoke’s theorem It relates a line integral to a surface integral in cases where curl of a vector exists in a vector field. It is given as If we consider the curl of a vector at each point in a vector field, and sum up all those curls within any chosen surface then, the sum will be equal to just the circulation of around the boundary of the chosen surface.  

12. Gauss’ law in electrostatics Consider a charge q and a closed surface surrounding it called Gaussian surface. The closed surface can be considered to be made up of a number of elementary surfaces each of area dS . If is the flux density at dS , then total electric flux over the entire surface is = q If there are a number of charges q 1 , q 2 , q 3 , …. inside the surface, then , Hence the total charge enclosed.  

13. Gauss’ law for magnetic fields & Maxwell’s equation in magnetostatistics Consider a closed surface in a magnetic field. Magnetic flux line always follows a closed loop irrespective of how the field is produced. For a closed surface in a magnetic field, for every flux line that enters into the surface, there must always be a flux line emerging out of the surface elsewhere. For entire closed surface, total outward flux = total inward flux Outward flux is taken as positive flux, inward flux is taken as negative flux. Therefore the total flux summed over the entire Gaussian surface = 0. In vector form, this aspect can be used to prove  One of the four Maxwell’s equation (where is the magnetic flux density).  

14. Ampere’s law Consider a point P in a magnetic field . Inorder to visualize the curl of the magnetic field, imagine a rectangular loop ABCD around P in a parallel to x-y plane. LetJ z and are the current density and unit vector respectively along z-direction. Finding the line integral of over the closed path ABCDA, it is possible to show that , where subscript-1 for curl H signifies that the curl is in a plane parallel to x-y plane.  

15. Similarly, for the curl in a plane parallel to y-z plane, we have Curl in a plane parallel to x-z plane is are the components of the curl of around the point P. is represented as Hence  Ampere’s circuital law (One of the Maxwell’s equation) where is the current density vector with components along the 3 cartesian coordinates.  

16. Biot-Savart’s law Biot-Savart’s law gives both the magnitude and direction of magnetic field intensity at a point due to the current in a differential element of a current carrying conductor. Let XY be a conductor carrying a current I. Consider a differential element of length dl at O on the conductor at which a tangent MN is drawn.

17. Let P be a point at a distance r from O, making an angle θ between MN and OP. As per Biot-Savart’s law, the magnitude of the magnetic field intensity at a point due to the current in the differential element is directly proportional to the product of the current I the magnitude of the length of the differential element dl sine of the angle between the tangent drawn to the element, and line joining the point and the element Inversely proportional to square of the distance between the point and the element  

18. Hence or , where K is proportionality constant = Therefore, The direction of at P as per the law is perpendicular to the plane containing the tangent drawn to the element and the line joining the point and the element. It is directed along the direction of progress of a right handed screw turned from dl to the line joining the element and the point through the smaller angle between them. In vector notation, Biot-Savart’s law is written as , where is directed along the direction of current and is the unit vector directed from the point O towards P  

19. Faraday’s law It can be stated as “the magnitude of the induced e.m.f . in a circuit is equal to the rate of change of the magnetic flux through it, and its direction opposes the flux change”. Induced emf e is expressed as , where ϕ is the flux linkage with the circuit such as a turn in a conducting coil. If there are N turns in the coil, then emf is expressed as  

20. Divergence of   Consider a point P in a region where there are charges. Let be the charge density at P. Due to the charges there will be electric field around P. It can be shown that Or (one of the Maxwell’s equation) is the divergence of  

21. Gauss divergence theorem The integral of the normal component of the flux density over any closed surface in an electric field is equal to the volume integral of the divergence of the flux throughout the space enclosed by the surface. It is represented mathematically as Consider a Gaussian surface in a region with certain charge density. Inside the surface, consider a differential volume element . Let be the charge within the element.  

22. If is the charge density, and since can vary continuously in the volume, we get Therefore If Q is the total charge enclosed by the Gaussian surface, then But, W.K.T, Therefore Applying Gauss’ law to Gaussian surface, From the above two equations, we get This is Gauss’s divergence theorem or divergence theorem.  

23. Equation of continuity For direct current in a circuit as shown in the figure, if we consider a closed surface at some part of it, then we see that the charge flow into the surface is equal to the charge outflow, which means there is no net charge within the surface i.e., It also represents Kirchoff’s current law according to which, the current inflow at any point in a circuit is equal to the current outflow from the same point. Considering the ampere’s law under time varying conditions for the field.  

24. The ampere’s law is expressed as Taking divergence on both sides, we get From vector analysis, divergence of curl for any field is always zero, hence  

25. This condition holds good for any electrical circuit with current flow under static conditions. But, it fails under time dependent variation to understand which, we can consider the following case. Consider an ac circuit with a capacitor Consider a closed surface which encloses only one plate of the capacitor. It is clear that, whenever there is any current flow into the closed surface, it is not accompanied by a simultaneous current outflow through any part of the surface (charges pile up on the capacitor’s plate). The reverse also holds good for the flow of current out of the surface. For to hold good, current inflow must be simultaneously equal to current outflow at any instant of time. Thus the condition fails in this case.  

26. Equation of continuity Consider a closed surface enclosing certain amount of charge. According to equation of continuity, if there is any charge outflow through this surface, it must be accompanied by a simultaneous reduction of charge within the surface. This can be represented vectorialy by applying the divergence theorem that  Equation of continuity. Where is the current density and is the charge density.  

27. Displacement current ( Modifictaion of Ampere’s law to suit the time varying condition) In order to make the ampere’s law to work under time varying field conditions, Maxwell suggested the following treatment We know from Gauss’s law that Differentiating w.r.t. time, we get Or W.K.T. from the equation of continuity that Using the above two equations, we get Or  

28. Now it is clear, that for the time varying case, it is not , but the equation must be considered, i.e., must be replaced by . Ampere’s circuital law in point form becomes for time varying field conditions. This equation is addressed as Maxwell-Ampere’s law. The quantity has the dimensions of the current density and was named by Maxwell as displacement current density.  

29. Maxwell’s equations From Gauss’s law in electrostatics, --------------------- (1) From Faraday’s law, ------------------------ (2) From Gauss’s law for magnetic field, -------------------(3) From Ampere’s law --------------------- (4) The above four equations are called as Maxwell’s equations.  

30. In case of static fields, the time dependent factors in the second and fourth equations vanish. Thus the list is modified as follows: ------------------------------- (1) -------------------------- (2) ------------------------------- (3) ---------------------------- (4)