EM fields and Current between 2 charged cylinders

[phantomvommand]

Homework Statement

Please see attached photos.

Relevant Equations

Ampere's Law

The task is to find the magnetic field between the 2 long cylinders, which extend to infinity. Integration is involved to find the total current passing through the Amperian Loop shown below. What I do not understand is why only sides 1 and 3 contribute to that B ds part of Ampere's Law. Isn't there magnetic field flowing parallel to sides 2 and 4 as well, like how a current in a straight wire creates a circular magnetic field that runs parallel to all sides of a loop around it?

I see this as similar to a straight wire, as there is only an X-component of current. The Y-component cancels out. However, there is "thickness" to this current.

 

[TSny]

The setup is not completely clear. Did you quote the complete problem statement exactly as it was given to you?

Do you have to find $\mathbf B$ for all points outside the two cylinders?

 

[phantomvommand]

The setup is not completely clear. Did you quote the complete problem statement exactly as it was given to you?

Do you have to find $\mathbf B$ for all points outside the two cylinders?

Apologies for leaving this out:
The 2 cylinders in the image above are immersed in a conducting fluid. They are thus effectively joined by a wire (which the current runs through), so there is no B-field outside of that region.

If it is still unclear:
Attached below is the full problem. I am only asking about the last part (8). However, I think you would still have to read through the above parts to understand the context; apologies that my summary was unclear.

Thanks for your help.

 

[TSny]

OK. Thanks for posting the entire problem statement.

Regarding your specific question about sides 2 and 4 of the Amperian path: First, consider any straight line parallel to the z-axis and lying outside the two cylinders. Let $a$ and $b$ be any two points on this line. Can you deduce anything about how the magnetic fields at these two points compare?

 

[phantomvommand]

OK. Thanks for posting the entire problem statement.

Regarding your specific question about sides 2 and 4 of the Amperian path: First, consider any straight line parallel to the z-axis and lying outside the two cylinders. Let $a$ and $b$ be any two points on this line. Can you deduce anything about how the magnetic fields at these two points compare?

The magnetic field there would only be in the z direction? But shouldn’t we be considering points lying on lines parallel to the y-axes, which is where paths 2 and 4 run parallel to?

 

[Steve4Physics]

Since you know sides 2 and 4 of the loop are being ignored, I'm guessing that you have an official solution, If so, maybe it would help if you included it.

At what point(s)/region(s) are you finding the magnetic field? Or are you being asked for a general expression for $\vec B (x,y,z)$? This is not clear from the question but if you have an official solution,it should be possible to tell.

 

[TSny]

The magnetic field there would only be in the z direction?

No. The field will not be in the $z$-direction. [EDIT: This is incorrect. $\mathbf B$ is in the positive or negative z-direction, except on the x-axis where B = 0.]

But if the cylinders are considered to be essentially infinitely long, you can say something about how the magnetic field varies as you move only in the $z$-direction. That is, does $\mathbf B$ depend on $z$? If so, how?

You should also be able to deduce from symmetry whether or not $\mathbf B$ has a non-zero $z$- component.

But shouldn’t we be considering points lying on lines parallel to the y-axes, which is where paths 2 and 4 run parallel to?

If you figure out how $\mathbf B$ depends on $z$, then consider a line parallel to the $z$-axis that passes through some point $a$ of path 2 and, therefore, also passes through some point $b$ of path 4. How does $\mathbf B$ compare for points $a$ and $b$? What does this tell you about how the line integral of $\mathbf B$ for path 2 compares to the line integral of $\mathbf B$ for path 4?

 

[phantomvommand]

Since you know sides 2 and 4 of the loop are being ignored, I'm guessing that you have an official solution, If so, maybe it would help if you included it.

At what point(s)/region(s) are you finding the magnetic field? Or are you being asked for a general expression for $\vec B (x,y,z)$? This is not clear from the question but if you have an official solution,it should be possible to tell.

It appears to me that there is only a Z-component of the B field, (See Eqn 30-31) but I am unclear why there is no Y-component along paths 2 and 4. The reason why $J_x$ = <that integral> is a result obtained from previous parts. I am mainly asking about the last line, when $B \dot l$ was replaced with $2B_zl$ directly, ignoring a possible $B_yl$ along paths 2 and 4.

 

[phantomvommand]

No. The field will not be in the $z$-direction.

But if the cylinders are considered to be essentially infinitely long, you can say something about how the magnetic field varies as you move only in the $z$-direction. That is, does $\mathbf B$ depend on $z$? If so, how?

You should also be able to deduce from symmetry whether or not $\mathbf B$ has a non-zero $z$- component.

If you figure out how $\mathbf B$ depends on $z$, then consider a line parallel to the $z$-axis that passes through some point $a$ of path 2 and, therefore, also passes through some point $b$ of path 4. How does $\mathbf B$ compare for points $a$ and $b$? What does this tell you about how the line integral of $\mathbf B$ for path 2 compares to the line integral of $\mathbf B$ for path 4?

Please see my above reply to Steve4Physics, it could be helpful. Because of symmetry, the Magnetic field in the Z direction along path 1 = Magnetic field in the Z direction along path 3. Furthermore, along Path 1/3, $B_z$ is constant as the wires are infinitely long.

it seems to me that the magnetic field through paths 2 and 4 are also entirely in the z direction, so B dot dz = 0.

 

[TSny]

Yes, you are right. I was mistaken in how I was thinking about the z-component of $\mathbf B$. I believe $B_z$ will be positive for positive values of $y$ and $B_z$ will be negative for negative values of $y$.

But here's how I was thinking about how to show that the line integrals of B for sides 2 and 4 cancel. Since the cylinders are infinitely long, the B-field cannot depend on the coordinate $z$. So, if you consider a point $a$ on side 2 and the corresponding point $b$ on side 4 that lies vertically below $a$, then $\mathbf B$ at points $a$ and $b$ are the same. So, the line integrals along 2 and 4 will be equal in magnitude. But they will have opposite signs because the directions of integration are opposite for the two sides.

Of course, if $\mathbf B$ is parallel to the z-axis everywhere, then, as you say, the line integrals for 2 and 4 are each equal to zero. This would be the easy way to see that sides 2 and 4 don't contribute.

 

[TSny]

It appears to me that there is only a Z-component of the B field, (See Eqn 30-31) but I am unclear why there is no Y-component along paths 2 and 4.

You can use the Biot-Savart law to show that $B_x = 0$ and $B_y = 0$ at all points.

Consider an arbitrary point $p$ where you want to calculate $\mathbf B$.

Consider the contribution to $\mathbf B$ at $p$ due to the current density at symmetrically placed points $a$ and $b$, as shown. Use the Biot-Savart law to show that the net x and y components of $\mathbf B$ at point $p$ due to these two current densities is zero.