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PY406 functions
创建日期：2024-05-06 17:09:17
所属分类：物理Physical
标签： PY406
functions

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PY406 Fall 


Special Instructions on this last homework: No collaboration is allowed. You may still
ask questions to the teaching staff, in person or on Piazza. Office hours will be held as
usual.
1. In HW10 we calculated the radiated power from the transition
2p (m=0)→ 1s
in hydrogen. Consider a much more complicated transition:
5p (m=0)→ 2s
The hydrogen wave functions (n,l,m) can be called by (example for 5p)
In[149]:= ResourceFunction["HydrogenWavefunction"][{5, 1, 0}, a, {r, θ, ϕ}]
Out[149]=
 1
a3
32 ⅇ- r5 a 2
5 π r 3750 a3-1125 a2 r+90 a r2-2 r3Cos[θ]
46 875 a
Calculate the electric dipole moment and resulting lifetime for this transition.
You may wish to check your answer. There are probably many places you can find
this. For example, see NIST Technical Note 1612 “Transitional Probabilities for One
electron Atoms” by Jonathan Baker (2008)
https://www.govinfo.gov/content/pkg/GOVPUB-
C13-22ce183612bcd5b2652e538177ee7805/pdf/GOVPUB-
C13-22ce183612bcd5b2652e538177ee7805.pdf
2. The earth is a magnetic dipole M that makes an angle ψ with the rotation axis. In
this problem we shall calculate the radiated power.
py406-s22-hw11.nb 1
Out[23]= rotation axis M
z/
ψ
The x and y components of M are changing with time. When viewed along the z axis
it looks like the dipole vector is undergoing circular motion as the earth rotates. This
circular motion can be viewed as the superposition of two one-dimensional oscillating
dipoles at right angles.
a) Determine a formula for the total radiated power from the 2 oscillating magnetic
dipoles. (Hint: you may want to consult how we calculated the power from a single
oscillating magnetic dipole.)
b) You travel down to the equator (taking some time off for a well-deserved vacation)
and measure the magnetic field to be 0.0005 T. Find the numerical value of M.
c) You measure the angle that M makes with the rotation axis and find ψ = 11∘ .
Calculate the total radiated power in W.
d) Compare your answer to power radiated by a neutron star with a radius of 10 km,
rotational period of 1 ms, and equatorial surface field of 108 T.
3. Consider a plane wave in frame S described by
E˜(z, t) = E0 ⅇⅈ kz-ωt x/
B˜ (z, t) = B0 ⅇⅈ kz-ωt y/
where as usual we physical fields are the real parts. In this problem, make use of the
fact that
 ωc , k
makes a 4-vector. Consider a frame S' that moves in the x direction with speed v,
relative to frame S.
a) Determine the frequency ω ' and wave vector k ' in S'.
2 py406-s22-hw11.nb
b) Find the electric and magnetic fields E˜ '(r ', t ') and B˜ ' (r ', t ') in S'.
c) Calculate E˜ '7 B˜ ' in S'.
d) Calcualte E˜ '× B˜ '* in S' and compare its direction to that of k ' as determined in a).
4. Consider two antennas that are far apart compared to both their size and wave-
length of operation. What follows does not depend on the antenna geometry. The
antennas each have an oscillating current.
Out[52]= J2J1
J1(r, t) = J1 (r) ⅇ-ⅈωt J2(r, t) = J2 (r) ⅇ-ⅈωt
Antenna 1 generates fields E1 (r, t) and B1 (r, t) while antenna 2 generates E2 (r, t)
and B2 (r, t).
a) Show that all the fields have the same time dependence, ⅇ-ⅈωt.
b) Show that in the radiation zone
E1*×B2 = E2*×B1
c) Show that it is generally true (in all space) that
μ0 E2* 7J1 - E1* 7J2 =∇ 7E1*×B2 - E2*×B1
d) Use the divergence theorem to show that
∫ ⅆ= ' E2* 7J1 = ∫ ⅆ= ' E1* 7J2
and reach the conclusion that the power transmitted into antenna 1 by antenna 2 is
the same as that transmitted into antenna 2 by antenna 1 and that therefore each
antenna works equally well as a transmitter or receiver.
5. A single charge e is at rest in Frame S at the origin. We are interested in the fields
at point P, position r = (x,y,z,).
a) In a Frame S’ that is moving with speed v in the x direction, use the formula
derived in HW10 to calculate the magnetic field at P.
B = 1-v  c 2qv×R)
4 π ε0 c 21-v  c 2 sinθ23/2 R 2
Give a numerical value for
py406-s22-hw11.nb 3
r = (1, 2, 3) μm v = 0.9 c
b) Calculate the magnetic field using the field transformation equations:
E 'x = Ex B'x = Bx
E 'y = γ Ey - vBz B'y = γ By + vc 2 Ez
E 'z = γ Ez + vBy B'z = γ Bz - vc 2 Ey
Show that this gives the same result.
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