Planetary Physics and Chemistry 2014: Homework 09

Planetary Physics and Chemistry 2014: Homework 09 - Magnetic fields & Magnetospheres
Due 25 November 2014
1. We want to show that the magnetic field at the center of a disc of thickness z and radius R
(with z << R) carrying an uniform electric current with linear density j is approximated by
B≈
µ0 jz ⎛ 1 z ⎞
⎜1 −
⎟
2 ⎝ 2 R ⎠
To do this, we consider a solenoid (i.e., a coil of wire) of length L and radius r and n turns/meter
of wire occupying a negligible thickness Δr and carrying a current I. The magnetic field on the
axis at one end of the solenoid is €
⎞
µ nI ⎛
r
B = 0 ⎜1 − 2 2 1/ 2 ⎟
2 ⎝ (r + L ) ⎠
Integrate the previous equation with respect to r, from 0 to R, replace L by z (the disc’s
thickness), and expand your result for the limit z << R.
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2. Let’s assume that the Earth’s magnetic field is dipolar and that the dipole aligns with the
Earth’s rotation axis. In spherical coordinates, the radial, latitudinal, and longitudinal
components of a dipolar magnetic field are given by
µ 2M
Br = − 0 3 sin θ
4π r
Bθ =
µ0 M
cos θ
4π r 3
€
Bϕ = 0
M is the dipolar moment, r is the distance to the center of the Earth, and θ is the latitude.
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(a) Calculate the amplitude B of the
€ magnetic field as a function of the latitude. Calculate the
amplitude Beq of the magnetic field at the surface of the Earth (r = R) along the equator.
Numerical application: M = 7.91×1022 A m2, R = 6371 km, µ0 = 4π×10-7 V s/(A m).
(b) Give an expression for B as a function of Beq, r, R, and θ.
(c) The magnetopause can be defined as the position where the internal pressure of the
magnetosphere Pm, balances the pressure of the solar wind, Pw. These pressures are given by
B2
Pm =
and Pw = ρv 2
2 µ0
where B is the Earth’s magnetic field amplitude, and ρ and v are the solar wind mass density and
velocity, respectively. Calculate the distance (from the center of the Earth) of the magnetopause
€
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Rm, as a function of R, Beq, and θ. Use the resulting equation to draw a plot showing how Rm
varies for different values of θ (for arbitrary values of Beq, ρ and v).
(d) Give an expression for the distance of the magnetopause in the equatorial plane, Rm,eq, as a
function of R and Beq. Numerical application (give results in terms of Earth’s radius):
(i) In quiet periods, the solar wind proton density is ρp = 6 protons/cm3, and the solar wind
velocity is v = 375 km/s (recall that mproton = 1.67×10-27 kg).
(ii) During a coronal mass ejection, the solar wind proton density is ρp = 12 protons/cm3, and
the solar wind velocity can reach up to v = 2000 km/s.
Compare the results from (i) and (ii), and comment: During which period is the magnetopause
more extended? Is this result surprising?
(e) The previous expression for Pm does not account for the pressure of the particles in the
plasmasphere, which is usually negligible (particle velocities are much slower there than in the
solar wind). What would be the effect of this additional pressure, if plasmasphere particles would
have higher velocities? Would it push the magnetopause away from the Earth, or pull it closer to
the Earth?
3. Consider a spherical conductor with a pattern of currents that gives a uniform axial magnetic
field throughout its volume. The field outside the sphere is equivalent to that of a dipole. Show
that the field energy outside the sphere is half of the field energy within the sphere. To do this,
note that the energy of a magnetic field (per unit volume) is given by B2/(2µ0). Assume that the
magnetic field inside the sphere has a uniform strength 2B0. Also assume that the external
strength as a function of radial distance r and colatitude θ is given by B = B0(1+3cos2θ)1/2 R3/r3,
where R is the radius of the sphere. Hint: You have to integrate over all θ and r outside the
sphere.

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