How many amperes are flowing in the Earth's core? To my surprise I couldn't find any estimate on how many amperes of electrical current are flowing in total in the Earth's core to be able to create the magnetic field strength $5\cdot 10^{-5}$ T on it's surface.
How can I estimate this electrical current?
How would the current be spatially distributed?
 A: We can do an estimate using Ampere's law
$$
\oint_A{\vec B}\cdot d{\vec l}~=~\int_A\vec j\cdot d\vec a.
$$
I am going to ball park numbers here. We think of the current along a wire at the outer core of the Earth with the wire representing an average current. The wire then has a radius between $1220$ km $3400$ km for the radius of the outer core. We set this at about $2000$ km. Now assume the magnetic field thorugh this loop is constant or we are concerned with the average so that
$$
2\pi|B|R~=~\pi|j|R^2,
$$
so the current density has magnitude $|J|~\simeq~2|B|/R$. NOw consider the current density as distributed in the inner core, where I think the currents are. This is a volume $V~=~\frac{4\pi}{3}(R_1^3~-~R_2^3)$ with $I~\simeq~|J|V$. Now put all this together and estimate $B~=~10^{-4}T$ and $V~\simeq~3.7\times 10^{18}m^3$ we then have as an approximation
$$
I~\simeq~2\times (10^{-4}T)(3.7\times 10^{18}m^3)/2\times 10^{6}m~=~3.7\times 10^8amps. 
$$
A: According to the dynamo theory,

[...] the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the Coriolis effect caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns (see also Taylor column) aligned with the rotation axis.

To compute the magnetic field, or the electric current density, one should then be able to solve the extremely complex nonlinear magnetohydrodynamic (MHD) equations for an electrically conducting fluid undergoing thermal convection in a rapidly rotating spherical shell. 
This is quite an ambitious goal: it is because of this that you couldn't find any estimate of the electric current flowing in the Earth's core (more exactly, it would be current density).
It must be mentioned, though, that there have been some important numerical studies, one of which was even able to predict the geomagnetic field reversal. 
A: Using the expression of a dipolar field, dimension of the Earth and dimension of the core:
$ B= \frac{\mu_0}{4 \pi} \frac{3 \mu.u - \mu}{r^3} $
with $\mu_0= {4 \pi} 10^{-7}$ uSI (international system units),
$r=6200$ km,
$B=4.6 \cdot 10^{-5} T
($u is a unit radial vector)
We obtain the magnetic moment of order
$\mu = 4.6\cdot 10^{-5}/10^{-7}*(6.2\cdot 10^6)^3 = 10^{23}$ uSI $= I S$
where S is the surface of the current loop.
From the inner core dimension of order 1000 km, the order of magnitude of S is $S=(10^6)^2=10^{12} m^2$ and the current is of order
$I=\mu/S= 4.6\cdot 10^{-5}/10^{-7}*(6.2\cdot 10^6)^3/(10^6)^2 = 10^{11} A$
The corresponding average current density is of order
$j = I/S = 4.6\cdot 10^{-5}/10^{-7}*(6.2\cdot 10^6)^3/(10^6)^4 = 0.1 A/m^2$
These are orders of magnitude.
