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
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...
Please Show Work Clearly. 12.3 Time-Harmonic Wave Equation. Using the source-free Maxwell's equations, show that a Helmholtz equation can be obtained in terms of the magnetic vector potential. Use the definition B = V X A and a simple medium (linear, isotropic, homogeneous material). Justify the choice of the divergence of A.
Show by substituting expressions for the dynamic potentials into Maxwell's equations that by setting the divergence of the vector potential A equal to the Lorentz gauge that 2 inhomogeneous wave equations result.
We use the last two Maxwell's Equations to derive the following two partial differential equations Use the above equations and the traveling wave equations for electric and magnetic field (E-Em Sin (Kx-ut) and B = B-Sin (Kx-ut) To prove speed of light formula C-1???.
Show Maxwell's equations in a semiconductor for a wavelength A.) Below B.) At and C.) Above the band gap of the a semiconductor. I'm very stuck on this, so insight on any part would be greatly appreciated.
Give a short answer to the following questions: wite down the Maxwell's equations in vacuum and name cach equation Deline magnetization in terms of magnetic dipoles in matter. What is its unit? What is Coulomb gauge? Use it to show the normal component of the vector potential must be continuous at the boundary of two materials Use one of the equations in (a) (which one?) to show that E is perpendicular to the wave vector k in a plane electromagnetic...
Problem 4: Time harmonic waves in lossy dielectric Start with Maxwell's equations and show that the electric field E(x, y, z, t) in a conductive material with conductivity σ satisfies the following wave equation a. 72 _ με.at? _ μσαί)F-0 b. Show that the following is a solution E(F, t)-(8 + 9) Eo e-kız cos(at-kez) where Eo is a constant and kR and k, are given by 0.5 w22 c. Obtain the direction of propagation for the wave in part...
9. (a)Using the Maxwell's equations prove that the wave equations for the electric and magnetic fields are given by 0t2 where l/c-μοεο
(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...