Energy and momentum flux (examples)

Problem:

(a)  Compare the intensity of a light bulb at a distance of 6 m from it to the intensity at 2 m from it.  Repeat this comparison for laser light.  Explain fully but briefly.
(b)  Compute the electric field corresponding to a focused light intensity of I = 1012W/cm2, and compare the result with the electric field experienced by the electron in a hydrogen atom.

Solution:

Problem:

A plane electromagnetic wave of intensity 6 W/m2 strikes a small pocket mirror of area 40 cm2 held perpendicular to the approaching wave.
(a)  What momentum does the wave transfer to the mirror each second?
(b)  Find the magnitude of the force that the wave exerts on the mirror.

Solution:

Problem:

A beam of light with an intensity of 108 W/m2 traveling in free space hits normally on a lossy mirror.  The mirror reflects 60% of the light and absorbs 40%.  What is the resulting radiation pressure on the surface of the mirror?

Solution:

Problem:

It has been proposed to drive a spacecraft remotely by directing an intense electromagnetic beam to the craft.
(a)  Is it more efficient to absorb the beam or reflect it from the craft?
(b)  If the beam carries 106 watts uniformly in an area of 5 m2 and the beam is reflected, how long would it take for a 1000 kg spaceship to reach a final velocity of 106 m/s, and how far would the craft travel in this time?

Solution:

Problem:

By considering the Poynting vector S = (1/μ0)(E×B) comment as quantitatively as possible on the energy flow in the following situations:
(a)  a long cylindrical wire of conductivity σ carrying a current I.
(b)  a stationary electric charge q sitting on top of a magnetic dipole with magnetic moment m.

Solution:

Problem:

An "air core" toroid of mean radius R and cross-sectional area πr2 is wound with N turns of wire.  (r << R).  A current with a time dependence I(t) = Kt is turned on at t = 0.
(a)  Find the energy stored in the magnetic field at time t.
(b)  Find the direction and magnitude of the Poynting vector at a point just inside the toroid at time t.
(c)  Using the Poynting vector, find the rate of change of the field energy inside the toroid at time t.  Check your answer with that of part (a).

Solution:

Problem:

A long coaxial cable consists of two concentric cylindrical conducting sheets of radii R1 and R2 respectively (R2 > R1).  The two conductors are connected to a battery which maintains a voltage V0 between them.  They carry equal and opposite currents I.
(a)  Use Ampere's law to calculate the magnetic field everywhere.
Plot the magnetic field as a function of r (use cylindrical coordinates).
(b)  Use Gauss' law to show that the electric field between the two conductors is
E
(r) = Cr/r2.  Express C in terms of R1, R2, and V0.  (Here r is a cylindrical coordinate.)
(c)  Calculate the Poynting vector.  Show that the power carried by the cable is P = V0I.

Solution:

Problem:

Consider two concentric spherical metal shells of radius a and b, a < b.  There is a charge +q on the inner and a charge -q on the outer sphere.  A magnetic dipole with dipole moment m is in the center of the inner sphere.  Find the angular momentum associated with the electromagnetic field of the system.

Solution:

Problem:

A (locally) plane electromagnetic wave in vacuum is propagating in the positive z-direction.  The angular frequency of the wave is ω.  The wave generator slowly decreases the amplitude of the wave.  At the position z = 0 the amplitude of the wave is E0(1 – at) for times between t = 0 and t = 1/a, with a << ω.
Consider a cylindrical surface as shown.

image

(a)  Find the average outward energy flux from the cylinder.
(b)  Find the field energy enclosed by the surface, and show that it decreases at a rate equal to the flux found in part (a).

Solution: