Calculate at home
• Derive continuity equation using fluid dynamics.
• Derive continuity equation using general transport equation (Bittencourt,
page 194).
• Write the Jacobian matrix and the Jacobian matrix determinant for
the transformation from spherical coordinates (r, θ, ϕ) to Cartesian
coordinates (x, y, z).
• Sir Edward Victor Appleton was awarded the Nobel Prize in physics in
1947 for his investigation of radio wave propagation in the ionosphere.
In his Nobel Lecture, he described the variations in the reception of
radio waves when the direct waves (near ground) interferred with waves
reflected from the ionosphere. Consider radio waves reflected from the
so-called E layer. The E-layer has a plasma density of 105 cm−3 and is
located at an altitude of about 100 km. What frequency radio waves
can reflect from the E-layer?
• Suppose a particle of mass m is moving in the force field of a onedimensional
harmonic oscillator. The force F (in x-direction ˆx) and
the potential energy U are given by
F = F(x) = −k x ˆx, U = U(x) =
1
2
k x2
The parameter k denotes a constant.
f1(x, v) = C1 exp −
m v2
2
− kx , f2(x, v) = C2 exp −
m v2
2
+ kx ,
f3(x, v) = C3 exp −
m v2
2
−
kx2
2
, f4(x, v) = C4 exp −
m v2
2
+
kx2
2
.
a) Only one of the following four functions solves the collisionless Boltzmann
kinetic equation for this case. Which one? Give a simple argument
or calculation.
b) From part (a) choose one of the three remaining functions and verify
that it does not solve the collisionless Boltzmann kinetic equation (a
few calculations are required).
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