In which direction is the magnetic field at point P?

[Meow12]

Homework Statement

Consider current in a coaxial cable as shown in cross section. The center conducting cylindrical wire carries current I out of the page, and the outer conducting cylindrical shell carries the same magnitude of current I into the page. Both currents are uniformly distributed across the cross-sectional area. In which direction is the magnetic field at point P?

Relevant Equations

Ampere's law: $\displaystyle\oint_C\vec{B}\cdot d\vec{l}=\mu_0I_{enc}$

Consider a circular Amperian loop oriented counterclockwise that is concentric with the circles in the figure and passes through P. By symmetry, $\vec{B}$ is everywhere tangent to this circular loop and has the same magnitude B everywhere on the circle.

By Ampere's law, $\displaystyle\oint_C\vec{B}\cdot d\vec{l}=\mu_0I_{enc}$

$\displaystyle\implies\oint_C Bdl=\mu_0I_{enc}$

$\implies B(2\pi r)=\mu_0I_{enc}$ where $r$ is the radius of the circular Amperian loop.

$I_{enc}$ is out of the page. Since the direction of integration is counterclockwise, by the right-hand rule, 'out of the page' is the positive current direction. So, $\vec{B}$ is in the same direction as the integration, i.e. counterclockwise. So, $\vec{B}$ points downward at P.

My question: I don't fully understand the symmetry argument (in red) about the direction of $\vec{B}$. For instance, I don't see why it cannot have a radial component.

 

[PhDeezNutz]

Look at the $I \vec{d \ell}$ that you have marked with an $X$ going into the page. Now draw $I \vec{d \ell}$ going into the page on the opposite side (bottom of the circle). Do the right hand rule for both currents at point $P$ and I think you will see the $x$ components cancel out (At the particular point of interest $y$ is the tangent and $x$ is the radial)

Does $x$ cancel out?

 

[kuruman]

What does the Biot-Savart law say about the direction of the magnetic field given a directed current element $~I~d\mathbf{l}$ and a position vector $\mathbf{r}$? Can that direction be radial in this case?

 

[Orodruin]

The same cylinder symmetry as for the infinite current carrying wire apply:

https://www.physicsforums.com/insights/symmetry-arguments-and-the-infinite-wire-with-a-current/

You don’t even need to refer to the Biot-Savart law - you just need to know that the magnetic field is a pseudo-vector.

 

[kuruman]

You don’t even need to refer to the Biot-Savart law - you just need to know that the magnetic field is a pseudo-vector.

Sure but at the introductory level, Biot-Savart is the first thing one is taught As an introduction to magnetic fields and their sources followed by Ampere's law. The transformation properties of pseudo-vectors , as elegant as the description may be, are not part of the introductory physics curriculum. The odds are in favor of OP knowing Biot-Savart but not pseudo-vectors. Of course, reading your insight article is a nice opportunity to learn about them and deepen one's understanding.

 

[Steve4Physics]

By symmetry, $\vec{B}$ is everywhere tangent to this circular loop and has the same magnitude B everywhere on the circle.
.
My question: I don't fully understand the symmetry argument (in red) about the direction of $\vec{B}$. For instance, I don't see why it cannot have a radial component.

I'd like to add this. The ‘symmetry argument’ (in red) is badly written. The wording implies that the (cylindrical) symmetry is the underlying reason for $\vec B$ being purely tangential. This is wrong. (The other posts address the real reason and hence why there is no radial component.)

But the part about $\vec B$ having the same magnitude everywhere on a (concentric) circle can sensibly be attributed to the cylindrical symmetry.

 

[Orodruin]

The wording implies that the (cylindrical) symmetry is the underlying reason for B→ being purely tangential.

This would depend on whether or not you consider the reflections part of the cylinder symmetry. The underlying reason for the field being tangential is the reflection symmetry in a plane containing the cylinder axis and the transformation properties of the magnetic field (pseudo-vector). The reflection is certainly a part of the symmetry group of the setup.

 

[Steve4Physics]

This would depend on whether or not you consider the reflections part of the cylinder symmetry. The underlying reason for the field being tangential is the reflection symmetry in a plane containing the cylinder axis and the transformation properties of the magnetic field (pseudo-vector). The reflection is certainly a part of the symmetry group of the setup.

No argument. But I wonder if that was the intended message of the Post #1 ‘red quote’. I suspect not!

 

[Orodruin]

But I wonder if that was the intended message of the Post #1 ‘red quote’. I suspect not!

"By symmetry" is a good guess if you have a feeling something is naturally the case but can't put your finger on it in the exam ...

 

[PhDeezNutz]

My grad school friends always joked that someone should stand up and proclaim “the answer is zero due to symmetry” regardless of what topic is being discussed at a weekly colloquium.

“Due to symmetry” is like the physicists version of “play free bird”

 

[Steve4Physics]

My grad school friends always joked that someone should stand up and proclaim “the answer is zero due to symmetry” regardless of what topic is being discussed at a weekly colloquium.

“Due to symmetry” is like the physicists version of “play free bird”

Spherical chicken in a vacuum.

 

[PhDeezNutz]

Spherical chicken in a vacuum.

I didn’t get it at first but now I do. Well played.

 

[Steve4Physics]

I didn’t get it at first but now I do. Well played.

Just in case you don't know, it's a reference. Google "spherical chicken in a vacuum." if required!