(e) Starting with the energy density of an electromagnetic field U = ε0E^2 /2 + B^2...
(e) Starting with the energy density of an electromagnetic field U = ε0E^2 /2 + B^2 /2µ0 and using two of Maxwell equations in free space, show that the rate of change of U is equal to the divergence of the Poyinting vector S, taken with the opposite sign, that is dU/dt = −∇ ∙ S
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(e) Starting with the energy density of an electromagnetic field
U = ε0E^2 /2 + B^2...
Electromagnetic waves transport energy. This problem shows you which parts of the energy are stored in...
Electromagnetic waves transport energy. This problem shows you which parts of the energy are stored in the electric and magnetic fields, respectively, and also makes a useful connection between the energy density of a plane electromagnetic wave and the Poynting vector. In this problem, we explore the properties of a plane electromagnetic wave traveling at the speed of light c along the x axis through vacuum. Its electric and magnetic field vectors are as follows: E = E, sin (kx...
Given the electric field phasor E-E) ρ-le-jkza, in cylindrical coordinates, where 1.4 , show that it represents an electromagnetic wave propagating in free space by using (a) Maxwell's equati...
Given the electric field phasor E-E) ρ-le-jkza, in cylindrical coordinates, where 1.4 , show that it represents an electromagnetic wave propagating in free space by using (a) Maxwell's equations, and (b) Helmholtz equation. (c) Find the magnetic field phasor H. 88 8-9 Given the electric field phasor E -(E.a, +jE a)e n free space, determine (a) propagation direction, (b) H, (c) & and (d) polarization state. 1th yeaeo circularly polarized waves.
Given the electric field phasor E-E) ρ-le-jkza, in cylindrical...
2) For an electromagnetic wave in free space having an electric field of amplitude E and...
2) For an electromagnetic wave in free space having an electric field of amplitude E and a magnetic field of amplitude B, the ratio of B/E is equal to A) C B) c2 C) 1/c D) 1/02 E) VC
In relativistic electrodynamics, the field tensor is given by 0 E, Ey E, с E. 0...
In relativistic electrodynamics, the field tensor is given by 0 E, Ey E, с E. 0 B2 - By FH = -B. 0 B. By -B. 0 (1) a) Write out the relativistic current four-vector JM in terms of the charge density p and the current density ). [4] b) Express the inhomogeneous Maxwell equations (the ones involving charge and current densities) in co-variant form using Fuv and JM. c) Show that your result from part b) recovers the inhomogeneous...
(b) (i) Starting with the definition of enthalpy, H = U + pl, and using a...
(b) (i) Starting with the definition of enthalpy, H = U + pl, and using a Maxwell relation, derive the following general equation of state. Write your derivation clearly and logically, showing all steps. You may use the following fundamental equation for change in internal energy without further proof: dU = Tds -pdv. TUDENT NAME NSHE # or My Nevadał: ii) Using the expression derived in (i) above, prove that for an ideal gas,
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL...
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL steps and justifications in your derivation. Using your result in (a) comment on how (b) the entropy behaves for an isothermal expansion of an ideal gas. Finally, show the validity of the following equations (c) U = F-TOOF) -T2 and at (T) 01 (d) C =-1(
1.) (a) State Maxwell’s equation for the curl of the magnetic and the electric field in...
1.) (a) State Maxwell’s equation for the curl of the magnetic and the electric field in free space. State the meaning of all the terms in the equations and identify the displacement current density. Using Maxwell’s equations, derive the wave equations for B. Show that the wave equations admit plane waves for the electric and magnetic fields in free space of the form ? = ??? ?(??−??) , ? = ??? ?(??−??) where ?? and ?? are constant vectors with...
As you work through the parts of this question you are going to show that the...
As you work through the parts of this question you are going to
show that the Maxwell equations naturally contain electromagnetic
waves. In a region of space that is void of all charges and
currents, ρ = 0 and J~ = 0 the Maxwell equations come out to
be:
Question 1: As you work through the parts of this question you are going to show that the Maxwell equations naturally contain electromagnetic waves. In a region of space that is...
(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...
Electromagnetic waves transport energy. This problem shows you which parts of the energy are stored in the electric and magnetic fields, respectively, and also makes a useful connection between the energy density of a plane electromagnetic wave and the Poynting vector. In this problem, we explore the properties of a plane electromagnetic wave traveling at the speed of light c along the x axis through vacuum. Its electric and magnetic field vectors are as follows: E = E, sin (kx...
Given the electric field phasor E-E) ρ-le-jkza, in cylindrical coordinates, where 1.4 , show that it represents an electromagnetic wave propagating in free space by using (a) Maxwell's equations, and (b) Helmholtz equation. (c) Find the magnetic field phasor H. 88 8-9 Given the electric field phasor E -(E.a, +jE a)e n free space, determine (a) propagation direction, (b) H, (c) & and (d) polarization state. 1th yeaeo circularly polarized waves.
Given the electric field phasor E-E) ρ-le-jkza, in cylindrical...
2) For an electromagnetic wave in free space having an electric field of amplitude E and a magnetic field of amplitude B, the ratio of B/E is equal to A) C B) c2 C) 1/c D) 1/02 E) VC
In relativistic electrodynamics, the field tensor is given by 0 E, Ey E, с E. 0 B2 - By FH = -B. 0 B. By -B. 0 (1) a) Write out the relativistic current four-vector JM in terms of the charge density p and the current density ). [4] b) Express the inhomogeneous Maxwell equations (the ones involving charge and current densities) in co-variant form using Fuv and JM. c) Show that your result from part b) recovers the inhomogeneous...
(b) (i) Starting with the definition of enthalpy, H = U + pl, and using a Maxwell relation, derive the following general equation of state. Write your derivation clearly and logically, showing all steps. You may use the following fundamental equation for change in internal energy without further proof: dU = Tds -pdv. TUDENT NAME NSHE # or My Nevadał: ii) Using the expression derived in (i) above, prove that for an ideal gas,
2. (25 pts) Derive the (a) Maxwell relation for the Helmholtz Free Energy F=U-TS. Show ALL steps and justifications in your derivation. Using your result in (a) comment on how (b) the entropy behaves for an isothermal expansion of an ideal gas. Finally, show the validity of the following equations (c) U = F-TOOF) -T2 and at (T) 01 (d) C =-1(
As you work through the parts of this question you are going to
show that the Maxwell equations naturally contain electromagnetic
waves. In a region of space that is void of all charges and
currents, ρ = 0 and J~ = 0 the Maxwell equations come out to
be:
Question 1: As you work through the parts of this question you are going to show that the Maxwell equations naturally contain electromagnetic waves. In a region of space that is...
(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...
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