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We use the last two Maxwell's Equations to derive the following two partial differential equations Use...
Use Maxwell's Equations to derive a decoupled set of wave equations for electric and magnetic fields...
Use Maxwell's Equations to derive a decoupled set of wave equations for electric and magnetic fields in a linear, homogeneous, isotropic media characterized by (µ, ε, σ) in the absence of sources. Then modify these equations to describe waves propagating in free space. Show all work, please
In this exercise we will derive the nonlinear interactions for the second harmonic using the scalar...
In this exercise we will derive the nonlinear interactions for the second harmonic using the scalar wave equation for the electric field (Eqn. 21.1-3) ot where ΕΞΕ(1,7) is the electric field. P-P(t, r)is the polarization n=n(a), cand Ho are the refractive index, vacuum speed of light and magnetic permeability. Mathematically, we can regard the wave equation as an inhomogeneous partial differential equation where the electric field wave is driven by the polarization of the media. The approximation we use to...
how did we get the following equation (1.9) from maxwells equations at e at where p is the density of free charges and j is the density of currents at a point where the electric and magnetic fie...
how
did we get the following equation (1.9) from maxwells
equations
at e at where p is the density of free charges and j is the density of currents at a point where the electric and magnetic fields are evaluated. The parameters and are constants that determine the property of the vacuum and are called the electric permittivity and magnetic permeability respectively The parameter c-1/olo and its numerical value is equal to the speed of light in vacuum,c 3 x...
Solve Use the Ampere-Maxwell Equation (the last of the 4 Maxwell equations) to derive the wave...
Solve
Use the Ampere-Maxwell Equation (the last of the 4 Maxwell equations) to derive the wave equation for the magnetic field, using a plane wave in a vacuum propagating in the x-direction, as shown in the figure. The Ampere loop to evaluate is shown as well. Note: this problem is very similar to the one derived in class today for the wave equation for the electric field dieve.The mpere oop to evalustes inavacum propagating in the diedrie eave equation for...
(a) Show that this field can satisfy Maxwell's equations if w and k are related in...
(a) Show that this field can satisfy Maxwell's equations if w
and k are related in a certain way.
(b) Suppose w=1010s-1 and
E0=1kV/m. What is the wavelength? What is the energy
density in joules per cubic meter, averaged over a large region?
From this calculate the power density, the energy flow in joules
per square meter per second.
(c) Show also that the electric field of associated with a
spherically symmetric wave may have the dependence Ei =
{Acos[k(r...
(a) Show that this field can satisfy Maxwell's equations if w and k are related in...
(a) Show that this field can satisfy Maxwell's equations if w
and k are related in a certain way.
(b) Suppose w=1010s-1 and
E0=1kV/m. What is the wavelength? What is the energy
density in joules per cubic meter, averaged over a large region?
From this calculate the power density, the energy flow in joules
per square meter per second.
(c) Show also that the electric field of associated with a
spherically symmetric wave may have the dependence Ei =
{Acos[k(r...
7. An EM wave is produced and has the following waves: E = Emax sin(kx-1200x10 st)...
7. An EM wave is produced and has the following waves: E = Emax sin(kx-1200x10 st) B= -2x10 sin(kx -t) in some direction a. find all the missing information in the electric and magnetic waves and rewrite them. b. What direction does the wave travel in? What direction is the magnetic field? c. What is the average intensity of the wave at its source? d. What kind of EM wave is this? How is it produced? e. The wave is...
Question 1. (a) Write down the differential form of Maxwell's equations in matter for the dynamic...
Question 1. (a) Write down the differential form of Maxwell's equations in matter for the dynamic case (where the electric and magnetic field can change with time), in the presence of free charges and currents. Describe all physical quantities and constants used. [10] (6) (b) Write down the integral form of Ampere's law in vacuum for the static (non time- dependent) case. Using Stokes' theorem, derive the differential form of Ampere's law. [4] (c) Two charges 91= 5 uC and...
Problem 3 Consider a possible solution to Maxwell's equations in vacuum given by A(x, t) =...
Problem 3 Consider a possible solution to Maxwell's equations in vacuum given by A(x, t) = Ao exp(i(kx - wt)), V(x, t) = 0 where A is the vector potential and V is the scalar potential. Suppose Ao, k and w are constants in space and time. a) Compute the time-dependent electric and magnetic fields from the given potentials. Show your work. b) Give the contraints, if any, on Ao, k and w imposed by the following two Maxwell's equations...
B(, t) = Bmar sin(kx - wt). (e) Using Faraday's law, find the electric field induced...
B(, t) = Bmar sin(kx - wt). (e) Using Faraday's law, find the electric field induced by the magnetic wave. (f) In part (e), what is the amplitude of the electric wave? Is there any phase difference between the electric wave and the magnetic wave? (g) The electric field you found in part (e) should also satisfy Ampére-Maxwell equation. Find the speed of the EM wave in terms of the constants en and yo using this requirement.
In this exercise we will derive the nonlinear interactions for the second harmonic using the scalar wave equation for the electric field (Eqn. 21.1-3) ot where ΕΞΕ(1,7) is the electric field. P-P(t, r)is the polarization n=n(a), cand Ho are the refractive index, vacuum speed of light and magnetic permeability. Mathematically, we can regard the wave equation as an inhomogeneous partial differential equation where the electric field wave is driven by the polarization of the media. The approximation we use to...
how
did we get the following equation (1.9) from maxwells
equations
at e at where p is the density of free charges and j is the density of currents at a point where the electric and magnetic fields are evaluated. The parameters and are constants that determine the property of the vacuum and are called the electric permittivity and magnetic permeability respectively The parameter c-1/olo and its numerical value is equal to the speed of light in vacuum,c 3 x...
Solve
Use the Ampere-Maxwell Equation (the last of the 4 Maxwell equations) to derive the wave equation for the magnetic field, using a plane wave in a vacuum propagating in the x-direction, as shown in the figure. The Ampere loop to evaluate is shown as well. Note: this problem is very similar to the one derived in class today for the wave equation for the electric field dieve.The mpere oop to evalustes inavacum propagating in the diedrie eave equation for...
(a) Show that this field can satisfy Maxwell's equations if w
and k are related in a certain way.
(b) Suppose w=1010s-1 and
E0=1kV/m. What is the wavelength? What is the energy
density in joules per cubic meter, averaged over a large region?
From this calculate the power density, the energy flow in joules
per square meter per second.
(c) Show also that the electric field of associated with a
spherically symmetric wave may have the dependence Ei =
{Acos[k(r...
(a) Show that this field can satisfy Maxwell's equations if w
and k are related in a certain way.
(b) Suppose w=1010s-1 and
E0=1kV/m. What is the wavelength? What is the energy
density in joules per cubic meter, averaged over a large region?
From this calculate the power density, the energy flow in joules
per square meter per second.
(c) Show also that the electric field of associated with a
spherically symmetric wave may have the dependence Ei =
{Acos[k(r...
7. An EM wave is produced and has the following waves: E = Emax sin(kx-1200x10 st) B= -2x10 sin(kx -t) in some direction a. find all the missing information in the electric and magnetic waves and rewrite them. b. What direction does the wave travel in? What direction is the magnetic field? c. What is the average intensity of the wave at its source? d. What kind of EM wave is this? How is it produced? e. The wave is...
Question 1. (a) Write down the differential form of Maxwell's equations in matter for the dynamic case (where the electric and magnetic field can change with time), in the presence of free charges and currents. Describe all physical quantities and constants used. [10] (6) (b) Write down the integral form of Ampere's law in vacuum for the static (non time- dependent) case. Using Stokes' theorem, derive the differential form of Ampere's law. [4] (c) Two charges 91= 5 uC and...
Problem 3 Consider a possible solution to Maxwell's equations in vacuum given by A(x, t) = Ao exp(i(kx - wt)), V(x, t) = 0 where A is the vector potential and V is the scalar potential. Suppose Ao, k and w are constants in space and time. a) Compute the time-dependent electric and magnetic fields from the given potentials. Show your work. b) Give the contraints, if any, on Ao, k and w imposed by the following two Maxwell's equations...
B(, t) = Bmar sin(kx - wt). (e) Using Faraday's law, find the electric field induced by the magnetic wave. (f) In part (e), what is the amplitude of the electric wave? Is there any phase difference between the electric wave and the magnetic wave? (g) The electric field you found in part (e) should also satisfy Ampére-Maxwell equation. Find the speed of the EM wave in terms of the constants en and yo using this requirement.
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