Biot-Savart Law & Magnetic Field: Clarification

[PFStudent]

Homework Statement

I had several questions about the magnetic field vector.

For example when applying the Biot-Savart Law, I was under the impression that each differential element ${d{\vec{s}}}$ containing a current ${I}$ along an infinitely long straight wire exerts a magnetic field according to the right-hand-rule (R.H.R.), ${\vec{B}}$ around and perpendicular to that differential segement ${d{\vec{s}}}$. As shown below,

However, I then came across this figure which more clearly shows that--from a differential segement ${d{\vec{s}}}$ with a current ${I}$ at a distance ${\vec{r}}$ oriented at an angle from ${d{\vec{s}}}$ there is a differential magnetic field ${d{\vec{B}}}$ due to this segment. As shown below,

Homework Equations

Biot-Savart Law

$${d{\vec{B}}} = {{\frac{{\mu}_{0}}{4{\pi}}}{\cdot}{\frac{Id{\vec{s}}{\times}{\vec{r}}}{{r}^{3}}}}$$

The Attempt at a Solution

This is a little confusing as I assumed that from the R.H.R. visualization of a magnetic field due to a current; that it showed that the magnetic field goes around the current at that particular segment (like a ring) and is oriented perpendicular to the current at that particular length segment.

So, then how is the magnetic field really supposed to be visualized if the R.H.R. visualization suggests it is exerted like a ring around a particular segment of current? As opposed to the other visualization suggesting that the magnetic field from a particular segement is like a ring that goes all the way up and down along the current and always perpendicular to the current.

So, how is it that the magnetic field is supposed to be visualized?

Any help is appreciated.

Thanks,

-PFStudent

 

[PFStudent]

Hey,

So, does anyone know how this works?

Thanks,

-PFStudent

 

[Astronuc]

${\vec{B}}$ = $\int{d{\vec{B}}}$. The magnetic field will not be radially symmetric if there is a kink in the wire.

 

[Galileo]

This is a little confusing as I assumed that from the R.H.R. visualization of a magnetic field due to a current; that it showed that the magnetic field goes around the current at that particular segment (like a ring) and is oriented perpendicular to the current at that particular length segment.

So, then how is the magnetic field really supposed to be visualized if the R.H.R. visualization suggests it is exerted like a ring around a particular segment of current? As opposed to the other visualization suggesting that the magnetic field from a particular segement is like a ring that goes all the way up and down along the current and always perpendicular to the current.

As the Bios-Savart law shows, there is a small contribution from each ds everywhere, but there is no component of B in the direction ds and r.

Both pictures are correct, so maybe I don't see where your difficulty lies. In both pictures is dB perpendicular to ds. Just imagine ds and r in one plane in the 2nd picture, then dB will point into the page.

Or is it that you thought the contribution was really a ring? (i.e. zero if r is not in the plane perpendicular to ds?) That's not true, as the law shows. The field drops off proportionally to 1/r^2. Remember that for an infinitely long wire, the field drops off as 1/r. That's because there are contributions from all parts of the wire. The greatest contribution does come from the part where r is perpendicular to ds.

 

[PFStudent]

Hey,

Or is it that you thought the contribution was really a ring? (i.e. zero if r is not in the plane perpendicular to ds?)

That is exactly what I thought, since I interpreted the R.H.R. visualization literally as an image--that the magnetic field existed only as a ring, in only the plane perpendicular to $${d{\vec{s}}}$$.

So, how would you best describe how the (total) magnetic field looks? Would you say that it is like a cylinder around the wire?

Thanks for the reply.

-PFStudent