Help with Ampere's Law: Toroid & Conducting Wire

[Oerg]

According to my textbook, the magnetic field of a toroid, according to ampere's law is given above as

$$\frac{\mu_0{NI}}{2\pi{r}}$$

but when i was looking through, I found that the magnetic field given by a piece of conducting wire is given as

$$\frac{\mu_0{I}}{2\pi{a}}(\cos{\theta_1}-\cos{\theta_2})$$

so that means amperes law must be given for an infinite length of current passing through an area bounded by the loop? If this is the case why does the formula work for the toroid? Is the formula just an approximation in this case where the height of the toroid is really large and the formula for the field is taken to be largely at the center?

 

[kof9595995]

so that means amperes law must be given for an infinite length of current passing through an area bounded by the loop?

No,Ampere's law works regardless of the shape of the circuit, but for a infinitely long wire it will give a very simple solution, for a toroid, the formula is an approximation, i think only when the wires are wounded tightly, I mean, no space between adjacent wires, the formula is accurate.

 

[confinement]

The solution for the toroid is exact, but only for the field at the exact center of the cross section.

 

[Oerg]

Ampere's law looks like a useless derivation of the biot savart law that is not really useful in application. I found myself resorting to the biot savart law to find the magnetic field in the toroid.

 

[Phrak]

Ampere's law looks like a useless derivation of the biot savart law that is not really useful in application. I found myself resorting to the biot savart law to find the magnetic field in the toroid.

Not at all. Ampere's circuital law including Maxwell's correction is quite general, one of Maxwell's equations, and dual of Faraday's law. Biot Savart has a reduced domain of applicability.

 

[Oerg]

Not at all. Ampere's circuital law including Maxwell's correction is quite general, one of Maxwell's equations, and dual of Faraday's law. Biot Savart has a reduced domain of applicability.

Hmm, are you saying that biot savart law is a result of Ampere's law? How can that be? My text derived Ampere's law from the magnetic field of an infinite wire with biot savart's law. It seems really weird to me that Ampere's law is more general than biot's law.

 

[confinement]

Hmm, are you saying that biot savart law is a result of Ampere's law? How can that be? My text derived Ampere's law from the magnetic field of an infinite wire with biot savart's law. It seems really weird to me that Ampere's law is more general than biot's law.

Are you also aware that Gauss' law is more fundamental than coloumbs law?

 

[Oerg]

Are you also aware that Gauss' law is more fundamental than coloumbs law?

nope, I have absolutely no idea and am bewildered. Isn't Gauss's law derived from Coulomb's law due to the scalar product of the electrostatic force with the area and so the integral of the small flux over the entire surface? If the electrostatic force is inversely proportionate to 1/r^3 instead of 1/r^2, Gauss's law would be invalid but if Gauss's law were changed, Gauss's law would be wrong and the electrostatic force would be valid.

Am I wrong? I think I am starting to get confused..

 

[confinement]

Logically, Coloumb's law and Gauss' law are equivalent. But Gauss' law is much more fruitful in terms of the physical and mathematical development of electromagnetism.