How Does Current Flow Create a Magnetic Field Around a Wire?

[v_pino]

Why will there be a magnetic field going around a wire that has current flowing?

I thought we need current and force perpendicular to each other in order to have a magnetic field. In a wire with current flowing, we don't have a force (motion).

 

[G01]

Yes there will be a magnetic field. You do not need any force to create a magnetic field.

For example a charge moving at a constant velocity will produce a magnetic field, even though the net force on the charge must be zero.

This all follows from Ampere's Law, which, for magnetostatics, is:

$$\oint \vec{B} \cdot d\vec{l} = \mu_o I_{enc}$$

In words this says that, the magnetic field going around an arbitrary path times the "perimeter" of that surface is equal to a constant times the current enclosed by the surface. So, the magnetic field only depends on the current, or moving charge, plus how far you are away from the current (contained in the perimeter term).