Download Induced electric current in the ocean

Induced electric current in the ocean
A fluid flows with uniform velocity v in the presence of a constant and uniform magnetic field B
perpendicular to v. The fluid has an electrical conductivity σ.
a) Find the electric current density J induced in the fluid.
b) Give a numerical estimate of |J| for the terrestrial oceans, knowing that the Earth’s magnetic field
has a typical value B ' 0.5 Gauss = 5 × 10−5 Tesla, the conductivity of sea water is σ ' 4 Ω−1 m−1
an a typical value of the flow velocity is v = 1 m/s.
c) Due to the appearance of the induced current the magnetic force tends to slow the flow. By
considering the force on a fluid element, estimate the time it would take for this effect to stop the
flow, if the magnetic force only was in action.
1
Solution
a) Due to the flow of the fluid, the charge carriers feel a force for unit charge equale to v × B, that
is equivalent to an electric field Eeq ≡ v × B. The induced current density is
J = σEeq = σv × B.
(1)
b) Inserting the typical values given in the text we obtain
J ' 4 × 1 × 5 × 10−5 A/m2 = 2 × 10−4 A/m2 .
c) For the fluid element we take a small cylinder with base surface δS and height
|δl|, with δl k J. The current intensity in the cylinder is I = JδS an the force is
thus given by F = Iδl × B = −BJδVv, where δV is the volume of the cylinder,
which in turn has a mass m = ρδV. The equation of motion, eliminating δV and
inserting J = σvB, is
dv
= −σB 2 v,
(3)
ρ
dt
whose solution is a decreasing exponential with a time constant
τ=
ρ
' 1011 s ' 3.5 × 103 yr,
σB 2
being ρ = 103 kg/m3 .
2
(2)
δl
δS
J
v
B
(4)