Why Must the Biot-Savart Law Be a Closed Curve?

[Petar Mali]

Definition of this law is given by

$$\vec{B}=\frac{\mu_0I}{4\pi}\oint \frac{d\vec{l}\times \vec{r}_0}{r^2}$$

why $$\oint$$ and no $$\int$$. Why must be closed curve?

Why not$$\vec{B}=\frac{\mu_0I}{4\pi}\int \frac{d\vec{l}\times \vec{r}_0}{r^2}$$

 

[xlines]

Definition of this law is given by

$$\vec{B}=\frac{\mu_0I}{4\pi}\oint \frac{d\vec{l}\times \vec{r}_0}{r^2}$$

why $$\oint$$ and no $$\int$$. Why must be closed curve?

Why not


$$\vec{B}=\frac{\mu_0I}{4\pi}\int \frac{d\vec{l}\times \vec{r}_0}{r^2}$$

If a conductor is not closed curve, what happens to current at the end of it? To integrate over finite, open curve - aren't you violating charge conservation law? What happens is that you have current along the curve and then it abruptly stops - what happens to flowing charges?

On the other hand, it happen fairly often to integrate over conductors that end up in infinity, so you can consider that as an example where you don't have to integrate over closed loop.

 

[Petar Mali]

This is in some textbooks used for conductors of finite length. But these are unrealistic objects in practice, if I understood you well?

 

[Born2bwire]

This is in some textbooks used for conductors of finite length. But these are unrealistic objects in practice, if I understood you well?

No, they are very realistic. Biot-Savart is valid for magnetostatics and thus all currents must be in closed loops since they are DC. If they weren't, then you would have an infinite build up of charges on one end and an infinite supply of charges at the other end of the unconnected wire.

 

[EHT]

If we are talking about instantaneous field, on the moment we have a steady current.
Actually we don't need a closed current loop for biot-savart, but ampere's law can only be used if $$\nabla . J = 0$$ everywhere

 

[Born2bwire]

If we are talking about instantaneous field, on the moment we have a steady current.
Actually we don't need a closed current loop for biot-savart, but ampere's law can only be used if $$\nabla . J = 0$$ everywhere

As I said in the other thread.

Biot-Savart is derived from Ampere's Law. Ampere's Law does not make any conditions upon the divergence of the current. Indeed, if it were, then we would be unable to find the Biot-Savart Law for an infinitesimal current element (which is equivalent to a point charge with constant velocity and thus the divergence of J is non-zero). When we talk about Ampere's Law, we are talking about the Ampere's Law as it appears in Maxwell's Equations.

And yes, we do need a closed current loop for Biot-Savart to make physical sense. We can of course derive the Biot-Savart Law for an infinitesimal current element but the proper physical implementation requires an implicit current loop for the reasons that I specified before. The Biot-Savart Law assumes magnetostatics and thus any current element of non-infinitesimal length is assumed to be providing a DC current indefinitely. If the current element was not a closed loop then this implies that we must have two separate objects that act as infinite sources and sinks of charges. The only cavaet I could see is if we have a discrete flow of charges (in which I guess we could say we have an infinitesimal countour) or if we have an infinite line current (in which we implicitly assume that there is a return wire set out at an infinite distance to close the loop).

If we have an AC current, we can easily imagine a non-loop current since the charges are oscillating along the wire (like in a monopole antenna) but then this is not a magnetostatic problem and thus Biot-Savart would not apply.

 

[EHT]

Biot-Savart is derived from Ampere's Law. Ampere's Law does not make any conditions upon the divergence of the current. Indeed, if it were, then we would be unable to find the Biot-Savart Law for an infinitesimal current element (which is equivalent to a point charge with constant velocity and thus the divergence of J is non-zero). When we talk about Ampere's Law, we are talking about the Ampere's Law as it appears in Maxwell's Equations.

Yes, Ampere's Law that I mentioned in my previous post is the one without maxwell's correction, therefore we need to add the effect of dispacement current due to the changing of charge density in both end of the wire to calculate the field

And yes, we do need a closed current loop for Biot-Savart to make physical sense. We can of course derive the Biot-Savart Law for an infinitesimal current element but the proper physical implementation requires an implicit current loop for the reasons that I specified before. The Biot-Savart Law assumes magnetostatics and thus any current element of non-infinitesimal length is assumed to be providing a DC current indefinitely. If the current element was not a closed loop then this implies that we must have two separate objects that act as infinite sources and sinks of charges. The only cavaet I could see is if we have a discrete flow of charges (in which I guess we could say we have an infinitesimal countour) or if we have an infinite line current (in which we implicitly assume that there is a return wire set out at an infinite distance to close the loop).

The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually. If we have a closed loop, biot-savart law can be used to calculate the field contribution of each random section of the loop.
If there is any nonzero divergence of J somewhere, we consider it as another contribution, which is due to the rate of change of electric field (due to the changing charge density)
thus the total magnetic field is the sum of these

sorry for my bad english

 

[rohannet]

Analogical Equation to E=kq/r²

 

[vorcil]

No, they are very realistic. Biot-Savart is valid for magnetostatics and thus all currents must be in closed loops since they are DC. If they weren't, then you would have an infinite build up of charges on one end and an infinite supply of charges at the other end of the unconnected wire.

Biot-savart holds for most practical things, including the current running through your house, which has a current that alternates 60 times every second (or more/less depending on which country you live in)

 

[Born2bwire]

Biot-savart holds for most practical things, including the current running through your house, which has a current that alternates 60 times every second (or more/less depending on which country you live in)

Technically it still requires a lot of cavaets and this still violates the exact physics. Biot-Savart is derived assuming magnetostatics. However, the low frequency of say 60 Hz can fall in the quasi-static region where we can still decouple the electric and magnetic fields. This can allow us to use Biot-Savart but only over electrically short lengths of loops. Once you start using it over length scales where the phase shifting becomes apparent then it becomes problematic. In addition, regardless of frequencies you still need the displacement current, which is not included in Biot-Savart, to properly account for a few things. The classic example is that of the gap in a parallel plate capacitor. I would also expect this to become a moot point if you calculate the fields numerically. The calculation of the magnetic field from currents exactly is not going to be any more computationally difficult than doing Biot-Savart.

 

[Selfluminous]

Biot Savart's law can be used to calculate magnetic field of a current element or part of a closed loop so i think in the general formula, the integral doesn't have to be on a closed current.