Ampère's law vs Biot Savart law

Question

So I feel like I understand how both these laws work however it seems like Ampère's law will find the strength of the magnetic field at a point (the point is taken as $z$ in this example and the red lines represent the contour of the magnetic field):

And the Biot-Savart law essentially finds the magnetic field at that point that is due to the current in the whole wire (where the red lines, in this case, represent the unit vector of the current at distances infinitesimally small in the wire):

Now my questions are, why does Ampère's law often apply to an infinitely long wire and the Biot-Savart Law apply to a short wire? Wouldn't it make sense for the Biot-Savart Law to apply for situations where there is a long wire since the current all the way through the wire would affect the magnetic field at a point $z$? And in that case, why would you ever use Ampère's law?

I feel Ampère's law would be innaccurate as it only represents the magnetic field directly perpendicular to the current wire at a certain length $dl$. But surely more of the wire would affect the magnetic field at point $z$ so how can you ever use Ampère's law? I feel that the Biot-Savart law makes more sense in every scenario, please help.

Answer 1

When you ask

why does Ampère's law often apply to an infinitely long wire and the Biot Savart Law apply to a short wire?

you're completely mistaken: both Ampère's law and the Biot-Savart law always hold.

More specifically, if you have a current $I$ running over a curve $\mathcal C_0$, then:

• The Biot-Savart law specifies the magnetic field $\mathbf B(\mathbf r)$ at any given position $\mathbf r$ in terms of an integral over the current-carrying circuit, $$\mathbf B(\mathbf r) = \frac{\mu_0}{4\pi} \int_{\mathcal C_0} \frac{I\mathrm d\mathbf l \times(\mathbf r-\mathbf r')}{\|\mathbf r-\mathbf r'\|^3}.$$

• Ampère's law specifies the circulation of $\mathbf B$ over any arbitrary curve $\mathcal C$ in terms of the current enclosed by said curve: $$\oint_\mathcal{C} \mathbf B\cdot\mathrm d\mathbf l = \mu_0 I_\mathrm{enc}.$$

If your goal is finding $\mathbf B(\mathbf r)$ at a given point, then you can use either or both to find it, and you normally use the simplest available route. If you have an infinite wire with lots of symmetry, the simplest route is to use Ampère's law, because you don't have to do any integrals. If you don't have such symmetry, you default to the Biot-Savart law, because then Ampère's law doesn't say anything about any individual point in space.

Ultimately, for magnetostatic calculations, the Biot-Savart law is probably the sturdiest way to obtain magnetic fields (though in some situations it can make sense to numerically solve Ampère's law in its differential form). However, in terms of fundamental importance, it is Ampère's law that wins the day, as part and parcel of the Maxwell equations, and therefore as a central part of the main framework of electrodynamics.

Answer 2

For starters, Ampere's law is just a slightly different variation of Maxwell's fourth equation, So it can be used almost anywhere(just like Gauss's law, but you will not get useful results except in some cases of evident symmetry.) Biot-Savart law is the magnetic analog of coulomb's law. You can find the field for any distribution of currents, provided you can integrate it to get useful results.

From Maxwell's 4th eqn, we have :

$\nabla \times \vec B = \mu_0. J + \frac{1}{c^2}.\frac{\partial \vec E}{\partial t}$

Amperes law neglected the second term, and in integral form looks like:

$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S}$

This one is familiar i suppose. One can apply amperes law to an infinite wire, only because $\vec B$ is uniform and circular about this wire. That is not the case for finite wires, where you cannot take $\vec B$ out of the integral on the left, and make good use of it. The law is not incorrect except in capacitor type cases when the second term in maxwell's eqn. needs to be taken into account. It will always give the correct answer. But without a certain degree of symmetry, example: a regular shape like a toroid,solenoid,circle etc, Ampere's law is generally useless. Biot-Savart law on the other hand, can give you B for any current distribution provided you can integrate suitably.