EM Waves Material PDF

UNIT-III ELECTROSTATIC WAVES Scalar Field: Scalar field is a field in two or three dimension which associates a real number at every point in space i.e. ϕ = ϕ(x, y, z) Example: 1. The temperature in a metal plate T...

UNIT-III ELECTROSTATIC WAVES Scalar Field: Scalar field is a field in two or three dimension which associates a real number at every point in space i.e. ϕ = ϕ(x, y, z) Example: 1. The temperature in a metal plate T(x, y) 2. Electrostatic potential near an electric field φ (x, y, z) 3. The gravitational potential V(x) Vector Field: Like scalar field we also have vector field in which a vector is associated with each point of space. Example: The intensity of electric field, gravitational field and magnetic field are example of vector field. Divergence of a vector field: ⃗ then divergence is defined to be, Given vector field ⃗𝑭 = 𝑷𝒊 + 𝑸𝒋 + 𝑹𝒌 𝜕𝑃 𝜕𝑄 𝜕𝑅 𝑑𝑖𝑣 𝐹 = ∇ ∙ 𝐹 = 𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧 Example:  2 2 2 Compute divF for F  x yî  xyzĵ  x y k̂ Solution: There really isn’t much to do here other than compute the divergence.  2   divF  x   x y   xyz   y z    x 2 y 2  2 xy  xz # We see that divergence basically indicates the amount of vector field F that is either converging to or diverging from a given point. # Generally the divergence of a vector field results in a scalar field (divergence) that is positive in some regions in space, negative in other regions and zero elsewhere. For example 1 Consider these vector fields in the region of a specific point 1  𝛁 ∙ ⃗𝑭 < 0 𝛁 ∙ ⃗𝑭 > 0 The field on the left is converging to a point, and therefore the divergence of the vector field at that point is negative. Conversely, the vector field on the right is diverging from a point. As a result the divergence of the vector field at that point is greater than zero. Example 2 When A is the velocity of a fluid at a point, then divA is the rate at which the fluid is diverging from that point per unit volume. If divA is positive at a point, then either the fluid is expanding and its density at that point is decreasing or the point is a source of fluid. If divA is negative then either the fluid is contracting, and its density id increasing at that point or the point is a sink. If divA is zero then there is no source or sink at that point nor its density changes i,e the fluid is incompressible. A vector which satisfy this condition is called as solenoidal. Example 3 In case of electric filed if the divergence at a point is positive it means there are positive charges at that point, if the divergence is negative it means the presence of negative charges at that point. Remember:  Gradient produces a vector field that indicates the change in the original scalar field, whereas;  Divergence produces a scalar field that indicates some change (i.e., divergence or convergence) of the original vector field. Note: 1. Divergence of a vector F is a Scalar. 2. If F is a conservative vector field then Curl F is zero 3. If F is solenoidal then Div F is zero. 4. For incompressible fluid divergence is zero Curl of a vector field: We define the curl of a vector fiel d𝐹 , written as 𝑐𝑢𝑟𝑙 𝐹 or ∇ × 𝐹 , as the cross product of operator Del with F. ⃗ then the curl is defined to be Given the vector field 𝐹 = 𝑃𝑖 + 𝑄𝑗 + 𝑅𝑘 𝑖 𝑗 𝑘 ⃗ 𝑐𝑢𝑟𝑙 𝐹 = ∇ × 𝐹 = | 𝜕 𝜕 𝜕 | 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑃 𝑄 𝑅 𝜕𝑅 𝜕𝑄 𝜕𝑃 𝜕𝑅 𝜕𝑄 𝜕𝑃 = ( − )𝑖 + ( − )𝑗 + ( − )𝑘 ⃗ 𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝑥 𝜕𝑥 𝜕𝑦 Note: 5. Curl of a vector F is a vector. 6. If F is a conservative vector field then Curl F is zero 7. If F is an irrotational then Curl F is zero. 2  8. Curl is a measurement of the circulation of vector field F(r) around point r. If a component of vector field F(r) is pointing in the direction dl at every point on contour Ci (i.e., tangential to the contour). Then the line integral, and thus the curl, will be positive. If, however a component of vector field F(r) points in the opposite direction - dl at every point on the contour, then curl at point r will be negative. 4. Likewise, these vector fields will result in a curl with zero value at point r. 5. Generally, the curl of a vector field result is in another vector field whose magnitude is positive in some regions of space, negative in other regions, and zero elsewhere. 6. For most physical problems, the curl of a vector field provides another vector field that indicates rotational sources of the original vector field. Significance of curl and divergence of a vector field A curl is in simplest word the calculation of the circulation per unit area. The physical significance of the curl of a vector field is the amount of “rotation” or angular moment of the stuffing of given region of space. It happens in fluid mechanics and elastic theory. It is also basic in the theory of electromagnetism. If the value of curl is zero then the field is said to be non rotational field. The divergence actually measure of how much the vector function is spreading out. Divergence of a vector field A is a measure of how much a vector field converges to or diverges from a given point. In simple terms it is a measure of the outgoingness of a vector field. Divergence of a vector field is positive if the vector diverges or spread out from a given point called source. Divergence of a vector field is negative if the vector field converge at that point called sink. If just as much of vector field points in as out the divergence is approximately zero. 3 Divergence and curl of Electric and magnetic fields Electric field is non-solenoidal, irrotational and conservative vector. Therefore, ∇.E  0, ∇xE = 0 Magnetic field is a solenoidal, rotational and non-conservative vector. Therefore, ∇. B = 0, ∇X B  0 Maxwell’s Equations: Maxwell’s equations are set of four equations that describe how electric and magnetic fields propagate, interact and how they are influenced by the objects. Maxwell’s equations in differential form are ⃗ ∙ 𝐸⃗ = 𝜌 Or ∇ ∇ ⃗ ∙𝐷⃗ =𝜌 𝜀 0 ⃗∇ ∙ 𝐵 ⃗ =0 ⃗∇ × 𝐸⃗ = − 𝜕𝐵 𝜕𝑡 ⃗ = 𝐽 + 𝜕𝐷 or ∇ ⃗ ×𝐻 ∇ ⃗ ×𝐵 ⃗ = 𝐽𝜇0 + 𝜇0 𝜕𝐷 𝜕𝑡 𝜕𝑡 Maxwell’s equations in Integral form 𝑞 1. ∫ 𝐸⃗ ∙ 𝑑𝑆 = ⁄𝜀0 or  D  ds =   dv or ∫𝑠 𝐷 ⃗ ⋅ 𝑑𝑆 = 𝑞 s v ⃗ 2. ∫ 𝐵 ∙ 𝑑𝑆 = 0 B  E  dl = -   ds 𝜕𝜙 3. ∫ 𝐸⃗ ∙ 𝑑𝑙 = − 𝜕𝑡𝐵 or c s t  D   H  dl =   J    ds 𝜕𝜙𝐵 ⃗ ∙ 𝑑𝑙 = 𝜇0 𝑖 + 𝜀0 𝜇0 4. ∫ 𝐵 or 𝜕𝑡 c s  dt  Proof: (Not required but yet to know and practice) ⃗ ∙ ⃗𝑬 = 𝝆 (Gauss’s law for electric field) 1. ⃗𝜵 𝜺 𝟎 𝑞 According to Gauss’s law in electrostatics 𝜙 = ∫𝑠 𝐸⃗ ⋅ 𝑑𝑆 = 𝜀 0 But 𝑞 = ∫𝑉 𝜌 𝑑𝑉 1 So, ∫𝑠 𝐸⃗ ⋅ 𝑑𝑆 = 𝜀 ∫𝑉 𝜌 𝑑𝑉 0 We also know that ∫𝑠 𝐸⃗ ⋅ 𝑑𝑆 = ∫𝑉 ∇ ⃗ ∙ 𝐸⃗ 𝑑𝑉 (According to Gauss Divergence theorem) 1 Now ∫𝑉 ⃗∇ ∙ 𝐸⃗ 𝑑𝑉 = 𝜀 ∫𝑉 𝜌 𝑑𝑉 0 𝜌 ⇒ ∇ ∙ 𝐸⃗ = ⃗ 𝜀0 ⃗ ⃗ ⃗⃗ 2. 𝜵 ∙ 𝑩 = 𝟎 (Gauss’s law in magnetic field) 4  In a magnetic field, the field B is always a dipole field. Because a magnetic monopole does not exist, even how small the volume is it contains equal number of north and south poles. Therefore the magnetic flux through a closed surface is always zero i.e. 𝜙𝐵 = ∫𝑠 𝐵⃗ ⋅ 𝑑𝑆=0 From divergence theorem we have, ⃗ ⋅ 𝑑𝑆 = ∫ ∇ ∫𝑠 𝐵 ⃗ ∙𝐵 ⃗ 𝑑𝑉 𝑉 So, ∫𝑉 ∇⃗ ∙𝐵 ⃗ 𝑑𝑉 = 0 ⇒ ⃗∇ ∙ 𝐵 ⃗ =0 𝝏𝑩 3. ⃗𝜵⃗ × ⃗𝑬 ⃗ =− (Faraday’s law) 𝝏𝒕 According to Faraday’s law of electromagnetic induction, 𝜕(𝑁𝜙𝐵 ) 𝜀 = ∫𝑠 𝐸⃗ ⋅ 𝑑𝑙 = − 𝜕𝑡 But ⃗ ⋅ 𝑑𝑆 𝑁𝜙𝐵 = ∫𝑠 𝐵 So we have ⃗ 𝑑𝐵 ∫𝑠 𝐸⃗ ⋅ 𝑑𝑙 = −∫𝑠 ⋅ 𝑑𝑆 𝑑𝑡 From Stokes theorem ∫𝑠 𝐸⃗ ⋅ 𝑑𝑙 = ∫𝑠 (∇ ⃗ × 𝐸⃗ ) 𝑑𝑆 So that, 𝜕𝐵⃗ ⃗ × 𝐸⃗ ) 𝑑𝑆 == −∫ ∫𝑠 (∇ ⋅ 𝑑𝑆 𝑠 𝜕𝑡 ⃗ 𝜕𝐵 ⇒ ⃗∇ × 𝐸⃗ = − 𝜕𝑡 ⃗⃗⃗ = 𝑱 + 𝝏𝑫 (Modified Ampere’s law in magnetic fields) ⃗⃗ × 𝑯 4. 𝜵 𝝏𝒕 According to Ampere’s law in magnetic fields ⃗∇ × 𝐻 ⃗ =𝐽 Where 𝐵 ⃗ = 𝐻 −𝑀 𝜇0 And J is the free current density in an infinitesimal area. Mean while if the electric field is not stable i.e. varying with respect to time and the variation of frequency is high enough and extended into radar frequency, then there will be another current density in 𝜕𝐷⃗ the medium known as the displacement current density ( ) where ⃗⃗⃗ 𝐷 is the 𝜕𝑡 displacement vector and is proportional to the variation of electric field (E) i.e. ⃗⃗⃗ 𝐷 ∝ 𝐸⃗ Or ⃗⃗⃗ 𝐷 = 𝜖𝐸⃗ , where 𝜖 is dielectric permittivity. The displacement current density does the same work as that of the conductive current density 𝐽⃗ ⃗ 𝜕𝐷 Thus there is another contribution along with 𝐽⃗ to introduce the magnetic field ⃗⃗⃗⃗ 𝐻 𝜕𝑡 5  ⃗ 𝜕𝐷 So that the total current density could be ⃗𝐽 + 𝜕𝑡 Hence the Ampere’s equation becomes to ⃗ 𝜕𝐷 ⃗ ×𝐻 ∇ ⃗ = 𝐽⃗ + 𝜕𝑡 # Units of D C/m2 E V/m B Tesla or V.s/m2 H A/m ρ C/m3 J A/m2 Physical significance of Maxwell’s equations: 1. According to Maxwell’s first equation the total electric flux through any closed surface is 1⁄ times the total charge enclosed by the closed surface. It is a steady state equation as 𝜀0 it does not dependent on time. Here for positive ρ divergence of electric field is positive and for negative ρ divergence is negative. It indicates that ρ is a scalar quantity. 2. The second equation represents the Gauss’s law in magneto statics. ⃗ ∙𝐵 ∇ ⃗ = 0 resulting that isolated magnetic poles or magnetic monopoles cannot exist as they appear only in pairs and there is no source or sink for magnetic lines of force. It is also independent of time i.e. steady state equation. 3. The third equation states that with time varying magnetic flux, electric field is produced in accordance with Faradays law of electromagnetic induction. This is a time dependent equation. 4. Fourth equation is a time dependent equation which represents the modified differential form of Ampere’s circuital law according to which magnetic field is produced due to combined effect of conduction current and displacement current density. Maxwell wave equations for free space Electromagnetic waves are transverse in nature. EM wave consists of electric vector and magnetic vector which vibrates perpendicular to each other and also perpendicular to the direction of the propagation of EM wave. In vacuum or free space EM waves travel with the speed equal to the speed of light and the expression for speed of EM wave in any medium is v = 1 1. In free space the equation of speed is v =   0 0 In order to understand the nature of waves, we consider free space, which is a large empty volume of space. Free space is a perfect dielectric and does not absorb (J=ρ = 0) waves. Let  = o and  = o for free space, using the Maxwell 3rd and 4th equations for free space (or a dielectric medium) 6  𝜕 2 𝐸⃗ ⇒ ⃗⃗⃗⃗ ∇2 𝐸⃗ − 𝜇𝑜 𝜀0 =0 𝜕𝑡 2 𝜕2 𝐸⃗ ⇒ ⃗⃗⃗⃗ ∇2 𝐸⃗ = 𝜇𝑜 𝜀0 𝜕𝑡 2 (1) This is the law that E must obey. A similar procedure for H gives us ⃗2 ⃗⃗⃗⃗ ∇2 𝐻 ⃗ = 𝜇𝑜 𝜀0 𝜕 𝐻2 (2) 𝜕𝑡 This is the law that H must obey. Equations (1) and (2) are very much similar to the general wave equation and constitute wave equations. These three dimensional vector wave equations describe the transmission of an electromagnetic wave through a uniform medium. The solutions of both equations have same form since the wave equations also have the same form. These equations lead to waves that can exist in free space. Even though the electric and magnetic fields of the waves start out on charges and currents, they detach themselves from them and move through free space as independent entities. The Poynting Theorem An electromagnetic wave carries energy with it as it propagates through space. The energy flow of electromagnetic wave is described by the Poynting vector P. It was proposed by J.H. Poynting. This theorem states that the cross product of electric field vector, E and magnetic field vector, H at any point is a measure of the rate of flow of electromagnetic energy per unit area or power flow per unit area at that point, that is P=ExH Let us consider the Maxwell’s curl equation ⃗ 𝜕𝐷 ⃗∇ × 𝐻 ⃗ = ⃗𝐽 + 𝜕𝑡 ⃗ ⃗ = 𝐸⃗. ⃗𝐽 + 𝐸⃗. 𝜕𝐷 Taking dot product with E , we have 𝐸⃗. ⃗∇ × 𝐻 (1) 𝜕𝑡 Now making use of the vector identity ⃗∇. (𝐸⃗ × 𝐻 ⃗)=𝐻 ⃗. ⃗∇ 𝑋 𝐸⃗ − 𝐸⃗. ⃗∇ 𝑋 𝐻 ⃗ 7  𝐸⃗. ⃗∇ × 𝐻 ⃗ =𝐻 ⃗. ⃗∇ 𝑋 𝐸⃗ − ⃗∇. (𝐸⃗ 𝑋 𝐻 ⃗) Therefore from equation (1) becomes ⃗ 𝜕𝐷 ⃗. ∇ 𝐻 ⃗ 𝑋 𝐸⃗ − ∇ ⃗. (𝐸⃗ × 𝐻 ⃗ ) = 𝐸⃗. 𝐽⃗ + 𝐸⃗. 𝜕𝑡 ⃗ ⃗ ⃗ 𝜕𝐵 − ∇  −𝐻 ⃗ ) = 𝐸⃗. 𝐽⃗ + 𝐸⃗. 𝜕𝐷 ⃗. (𝐸⃗ × 𝐻 𝜕𝑡 𝜕𝑡 ⃗ ⃗ ⃗ 𝜕𝐻 − ⃗∇. (𝐸⃗ × 𝐻  − 𝐻 ⃗ ) = 𝐸⃗. ⃗𝐽 + 𝐸⃗. 𝜕𝐷 Since B = μH 𝜕𝑡 𝜕𝑡 ⃗ ⃗ ⃗ ) = 𝐸⃗. ⃗𝐽 + 𝐸⃗. 𝜕𝐸 + 𝐻 ⃗. (𝐸⃗ × 𝐻  −∇ ⃗ 𝜕𝐻 Since D = ϵE 𝜕𝑡 𝜕𝑡 𝜕𝐸⃗ 𝜖 𝜕𝐸⃗ 2 𝜕 𝐸⃗ 2 As 𝐸⃗. 𝜕𝑡 = 2 𝜕𝑡 = 𝜕𝑡 ( 2 ) ⃗ 𝜕𝐻  𝜕𝐻 ⃗2 𝜕 𝐻 ⃗2 and 𝐻 ⃗ 𝜕𝑡 =2 𝜕𝑡 = 𝜕𝑡 ( 2 ) ⃗ ) = 𝐸⃗. ⃗𝐽 + 𝜕 (𝐸 + 𝐻 ) ⃗2 ⃗2 Hence, ⃗. (𝐸⃗ × 𝐻 −∇ 𝜕𝑡 2 2 Rearranging the terms and integrating it over a volume V, we obtain 𝜕 𝐸 𝐻 ⃗2 ⃗2 ∫𝑣 𝐸⃗. 𝐽⃗ 𝑑𝑣 = − 𝜕𝑡 ∫𝑣 ( 2 + 2 ) 𝑑𝑣 − ∫𝑣 ∇ ⃗. (𝐸⃗ × 𝐻 ⃗ ) 𝑑𝑣 Using the divergence theorem the last term can be changed from volume integral to a surface integral. Thus, ⃗. (𝐸⃗ × 𝐻 ∫ ∇ ⃗ ) 𝑑𝑣 = ∫ (𝐸⃗ × 𝐻 ⃗ ). 𝑑𝑠 𝑣 𝑠 𝜕 𝐸⃗ 2 𝐻 ⃗2 ∫ 𝐸⃗. ⃗𝐽 𝑑𝑣 = − ∫( + ) 𝑑𝑣 − ∫ (𝐸⃗ × 𝐻 ⃗ ). 𝑑𝑠 𝜕𝑡 2 2 𝑠 𝑣 𝑣 This is pointing theorem. Significance of terms The rate of dissipation of energy in the volume V = the rate at which the stored energy in V is decreasing plus the rate at which energy is entering the volume from outside. ∫𝑣 𝐸⃗. ⃗𝐽 𝑑𝑣 is a generalization of Joule’s law. It represents the instantaneous power dissipated in volume V. 8 𝜕 𝐸⃗ 2 𝐻 ⃗2 ∫ ( 𝜕𝑡 𝑣 2 + 2 ) 𝑑𝑣 is negative time derivative of the stored electric and magnetic energy in volume. Negative sign indicates the power being delivered by the field inside the volume to the region outside the volume. It represents the rate at which the stored energy in the volume is decreasing. ∫𝑠 (𝐸⃗ × 𝐻 ⃗ ). 𝑑𝑠 is from application of the Principle of conservation of energy. A negative value represents the rate of flow of energy inward through the surface of the volume and positive value represents the rate of flow of energy outward through the surface enclosing the volume. 9