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ELECTRODYNAMICS MAXWELL EQUATIONS & ELECTROMAGNETIC WAVES  Scalar and Vector Field  A field is a spatial distribution of a quantity; in general, it can be either scalar or vector in nature.  Region in space, every point of which is characterized by a scalar quantity is known as sca...

ELECTRODYNAMICS MAXWELL EQUATIONS & ELECTROMAGNETIC WAVES  Scalar and Vector Field  A field is a spatial distribution of a quantity; in general, it can be either scalar or vector in nature.  Region in space, every point of which is characterized by a scalar quantity is known as scalar field.  An example of a scalar field in electromagnetism is the electric potential. Other examples include temperature field, pressure field , gravitational potential etc.  Region in space, each point of which is characterized by a vector quantity is known as vector field.  Examples of vector field are electric field, gravitation field, magnetic field , magnetic potential etc.  Del Operator  The operators are mathematical tools or prescriptions. The operators have no direct physical meaning. However, they acquire significance when operated upon another function.  The del operator is the vector differential operator,  Represented by  Note: DEL operator is not a vector quantity in itself, but it may operate on various scalar or vector fields. Gradient  Physical Significance of Gradient The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector that points in the direction of greatest increase of a function  Physical Significance ??? Thus the rate of change of Φ in the direction of a unit vector a is the component of grad Φ in the direction of a (i.e. the projection of grad Φ onto a ). The maximum value of the directional derivative occurs when the directional vector a coincides with the direction of grad Φ. Thus the directional derivative achieves its maximum in the direction of the normal to the level surface Φ(x, y, z) = c at P.  Then small change in scalar field as we alter all three variables by small amount dx, dy and dz is given by fundamental theorem of partial derivative, i.e. Divergence Physical Significance  Divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point  The divergence of vector field A is defined as the net outward flux per unit volume over a closed surface S.  The div. A at a point is measure of how much the vector A spread outs. If Divergence of vector field is zero , then it is also termed Solenoidal Field  Curl Physical Significance of Curl: The maximum value of the circulation density evaluated at a point in the vector field is known as curl of vector field The rotation with maximum value is known as curl and is a vector quantity. Thus curl of vector field signifies the whirling nature or circulation of the vector field (A) around any point O. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation Conservative Fields  For a conservative vector field , CURL IS ZERO  The curl of a vector field is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point.  The magnitude of Curl is the limiting value of circulation per unit area.  Curl is simply the circulation per unit area, circulation density, or rate of rotation (amount of twisting at a single point).  To be technical, curl is a vector, which means it has a both a magnitude and a direction. The magnitude is simply the amount of twisting force at a point.  The direction is a little more tricky: it's the orientation of the axis of your paddlewheel in order to get maximum rotation. In other words, it is the direction which will give you the most "free work" from the field. Imagine putting your paddlewheel sideways in the whirlpool - it wouldn't turn at all. If you put it in the proper direction, it begins turning. Gauss Divergence Theorem  This Theorem helps to transform a surface integral into volume integral.  It states that the surface integral of any vector field through a closed surface is equal to volume integral of the divergence of vector field taken over the volume enclosed by the closed surface. Mathematically,  Numerical Given that r⃗ is a position vector. Using Gauss’s divergence theorem find the value of ∯ 𝑟⃗. 𝑑𝑠⃗. Stoke’s Theorem  It states that line integral of the tangential component of a vector field A over a closed path is equal to the surface integral of the normal component of the curl A on the surface enclosed by path.  Mathematically,  Numerical If r⃗ is a position vector of a point in space, then prove that ∮ 𝑟⃗. 𝑑𝑟⃗ = 0  Numerical  Prove that 𝐴⃗ = 𝑦𝑧𝑖ˆ + 𝑥𝑧𝑗ˆ + 𝑥𝑦𝑘ˆ is both irrotational and solenoidal  If F=3x^2 y-y^ 3 z^ 2 , find the value of gradient of the function F at point (1, -2, -1).  Green’s Theorem  Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C.  If L and M are functions of (x, y) defined on an open region containing D and having continuous partial derivatives there, then  Mathematically M L C Ldx  Mdy D ( x  y )dxdy * Note: Here curve C has a positive orientation if it is traced out in a counter- clockwise direction Continuity Equation  Maxwell’s equations Maxwell's equations are a set of James Clerk Maxwell, one of the partial differential equations that, world's greatest physicists, was together with the Lorentz force law, Professor of Natural Philosophy at form the foundation of classical King's from 1860 to 1865. It was electrodynamics, classical optics, and during this period that he electric circuits. These fields in turn demonstrated that magnetism, electricity and light were different underlie modern electrical and manifestations of the same communications technologies. fundamental laws, and described all Maxwell's equations describe how these, as well as radio waves, radar, electric and magnetic fields are and radiant heat, through his unique generated and altered by each other and elegant system of equations. and by charges and currents. They are These calculations were crucial to named after the physicist and Albert Einstein in his production of mathematician James Clerk Maxwell, the theory of relativity 40 years later, who published an early form of those and led Einstein to comment that 'One scientific epoch ended and equations between 1861 and 1862. another began with James Clerk Maxwell'. Maxwell’s equations  Maxwell's four equations describe the electric and magnetic fields arising from distributions of electric charges and currents, and how those fields change in time.  They were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents, plus the profound intuition of Michael Faraday.  Maxwell's own contribution to these equations is just the last term of the last equation -- but the addition of that term had dramatic consequences. It made evident for the first time that varying electric and magnetic fields could feed off each other -- these fields could propagate indefinitely through space, far from the varying charges and currents where they originated.  Previously these fields had been envisioned as tethered to the charges and currents giving rise to them. Maxwell's new term (called the displacement current) freed them to move through space in a self-sustaining fashion, and even predicted their velocity -- it was the velocity of light! Maxwell’s equations Maxwell 1st Equation: Significance Maxwell 2nd Equation: Maxwell 3rd Equation: According to Faraday’s law, is the Magnetic Flux within a circuit, and EMF is the electro- motive force Significance of Maxwell’s third equation (i) It summarizes the Faraday’s Also, law of electromagnetic induction. (ii) This equation relates the space variation of electric field with time variation of magnetic field (iii) It is time dependent differential equation. (iv) It proves that the electric field can be generated by change in magnetic field Maxwell 4th Equation: According to Ampere’s Law Maxwell realized that the definition of the total current density is incomplete and suggested to add another term Significance of Maxwell’s fourth equation (i) It summarizes the modified form of Ampere’s ciruital law. (ii) It is time dependent differential equation. (iii) Maxwell’s fourth equation relates the space variation of magnetic field with time variation of electric field (iv) It also proves that magnetic field can be generated by changing electric field PROPAGATION OF ELECTROMAGNETIC WAVE IN FREE SPACE The Maxwell’s equation for free space (=0 and J=0) can be written as What , why & How??????  Define : Curl, Divergence &  Derive differential Maxwell Gradient equations. Also write their  Explain the physical significance: physical significances. Curl, Divergence & Gradient  Derive the equations for  Write the expression for Del- electromagnetic wave operator propagation in free space using Maxwell equations, and hence  Write continuity equation and its calculate the value of c (velocity physical significance of light).  Derive an equation which  Write Stoke’s and Divergence’ express conservation of charge theorems. in a localized volume.  What are the conditions for  Write Maxwell equations in both irrotational, solenoidal and differential and integral form conservative fields, resp.?