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Universidad Nacional de Colombia

Electrodinamica II

Ejercicios: Lista 4Profesor Carlos Viviescas

Fecha de entrega: 17.09.14

Exercise 10 (3 points) Wave packet

General solutions of the wave equation can be written as superpositions of plane waves. In one dimen-sion, its general form is

(r, t) =

dk

2a(k) ei(kxt) , (1)

with dispersion relationk = w/c.

One then talks about a wave packet center in k0 if the distribution a(k) shows a sharp peak in k0 anddecays for values ofk away from the center (see the figure). The width of the wave k is the mean

k

k0k

ak

Paquete de onda centrado en k0 y de ancho k

squared root deviation (standard deviation) respect to the meank of the intensity|a(k)|2

,

(k)2 =

+

dk (k k)2|a(k)|2

+

dk |a(k)|2,

in which

k =

+

dk k|a(k)|2

+

dk |a(k)|2

.

1. For (1), show the uncertainty relation x k 12 . Assume for simplicity that x = 0 and k = 0.Explain the implications of this uncertainty relation.

Hint: Use Schwarz inequality.

+

dx |g(x)|2+

dx |h(x)|2 14

+

dx [g(x)h(x) +g(x)h(x)]

2

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valid for well behaved functionsg(x) yh(x), and the relation

+

dx

(x, t)x2

=

+

dk

2k2|a(k)|2

that is obtained as a corollary of Percival theorem.

2. Argument in a clear way (without repeating the calculation) the validity of the uncertainty relationt 12 , for the wave packet (1). ( is commonly call the band width of the wave spectrum|a()|2.) Explain the implications of this uncertainty relation.

3. The ultra-short pulses use in femto-chemistry to observed chemical reactions in the time scale inwhich they occur have t 1013 s. For these pulses:(a) Find the minimum possible band width .

(b) Find the minimum possible value of x. This last quantity gives an approximation to thesize of the wave packet in space.

Hint: Use the dispersion relationk = /c to determinek.

4. Wave packets for which x k = 12 are called packets of minimum uncertainty. Show that theGaussian wave packet center at the origin and with amplitud

a(k) = 1

2ke

k2

4(k)2 ,

is a minimum uncertainty wave packet.

Exercise 10 (3 points) Spherical Waves I

Consider the following electric field

E(r,,,t) = Asin()

r

cos(kr t) 1

krsin(kr t)

,

with the dispersion relation k

=c. (This is the simplest form of a spherical wave. To simplify notationcall (kr

t) = u in your calculations.

1. Show that E satisfies Maxwell equations in vacuum and find the associated magnetic field.

2. Evaluate the Pointing vectorS. The time average ofS over a cycle is equal to the intensity vectorI S and gives the power per surface unit that the wave carries.

3. Calculate the flux ofI through a spherical surface in order to find the total radiated power.

Exercise 11 (3 points) Spherical Waves II

The wave equation in three dimensions is given by

2= 1c2

2

t2 .

Taking now a point source localized at the origin which emits waves with spherical symmetry:

1. Show that the emitted waves are of the form

=A

rei(krt) .

Hint: due to its spherical symmetry the wave function should only depend onr, therefore

2= 1r2

r

r2

r

=

1

c22

t2 .

Consider the substitution = r to solve the equation.

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Assume now that the source is a point dipole in direction ez.

2. Show that

E= ( ez)

is a solenoidal solution (

E= 0) of the wave equation

2E= 1c2

2E

t2 .

3. Find the magnetic associated field

Hint: use Bt

= iB (where does this relation come from?)4. Write the fields explicitly using spherical coordinates and show that in the limits

kr 1 (near field)

E=2p(t)cos

40r3 er+

p(t)sin

40r3 e

B= 0

corresponding to a quasi-static electric dipole oscillating with momentum p p0eitez =40Ae

it ez.

kr 1 (far field)

E= k2A sin

r e

B= k2A sin

cr e

here 1/r is typical for fields generated by radiation.

5. In order to see the radiative character of the far field, evaluate its Poynting vector and show thatthe power through the spherical surface is independent of its radius.

Hint: since we are dealing with time oscillating fields consider the time average of the Poyntingvector.

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