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It seems a little counter intuitive that neutron stars possess such strong magnetic fields. Its electric charge is presumably zero, so however fast it spins, it shouldn't generate any magnetic field. Or is it due to the electric charges of quarks or their intrinsic spins?

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The strong magnetic fields in neutron stars are supposed to come from magnetic flux conservation. If we have:

$\Phi_B = \int B\ dS = const$

where $\Phi_B$ is the magnetic field flux, $B$ is the magnetic field strength, and $dS$ is the elemental closed surface; then, this integral is constant through the surface.

If we consider the star surface over which take the integral, than

$S = 4\pi R^2$

where $R$ is the star radius. This can be translated, altogether with the magnetic flux conservation law, as:

$B_f = B_i (\frac{R_i}{R_f})^2$

where $i$ and $f$ are the indices for initial and final stages. We know that the star implodes from a whatever star size to $\sim10$ km. So the radii ratio is huge. You just need a starting magnetic field of $10-100$ G, to get a final magnetic field of the order of $10^{12}$ G, that is typical in neutron stars.

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