Confusion on the magnitude of magnetic fields

[Gourab_chill]

Homework Statement

I've put question below.

Relevant Equations

B=µNI; N=no of turns for unit length of solenoid

Here, the correct options are A,D.
Solution:

I got A as answer as ∫ B.dl=µI. But, the answer to the question says that it is a solenoid and therefore Bx=0 for point P. Here I'm a bit confused. I know this system resembles a solenoid in some ways, then By must have some finite value, but ampere's law says there should be no magnetic field in the pipe or it's magnitude is zero. Where am I mistaking?

 

[Delta2]

I believe you are applying Ampere's law wrongly in the interior of the pipe (probably because you are misconceiving the symmetries present in this system). From Ampere's law we can't deduce that the B-field will be zero inside, not for this system. If the pipe was doing only translational motion then we could deduce that the B-field would be zero inside but because it does rotational motion too around the y-axis, it generates a component ##B_y## in the interior that it isn't zero.

 

[Gourab_chill]

I believe you are applying Ampere's law wrongly in the interior of the pipe (probably because you are misconceiving the symmetries present in this system). From Ampere's law we can't deduce that the B-field will be zero inside, not for this system. If the pipe was doing only translational motion then we could deduce that the B-field would be zero inside but because it does rotational motion too, it generates a component ##B_y## in the interior that it isn't zero.

So, where is exactly ampere's law applicable? If the current is moving around the pipe (due to rotational motion as here), we cannot apply it? Does the current have to be perpendicular to the imaginary circle we consider in ampere's law?

 

[Delta2]

Ampere's law is applicable everywhere, I just meant that you apply the law in the wrong way for this system.
Please write a post and explain with as much detail as possible how do you apply Ampere's law for this system so that you infer that ##B_y=B_x=B_z=0## for the interior of the pipe. ##B_x## and ##B_z## are indeed zero but not ##B_y##.

 

[Gourab_chill]

Ampere's law is applicable everywhere, I just meant that you apply the law in the wrong way for this system.
Please write a post and explain with as much detail as possible how do you apply Ampere's law for this system so that you infer that ##B_y=B_x=B_z=0## for the interior of the pipe. ##B_x## and ##B_z## are indeed zero but not ##B_y##.

All I knew till now is that by applying ampere's law the net magnetic field at a point is zero if it does not encloses any current.
For solenoids as I looked up, the derivation also uses ampere's law, however here the loop encloses current due to it's position which I've tried to show in the first figure. What if we take the loop as in the second figure? The magnetic field should be zero in the second case or is the loop position wrong?

 

[Delta2]

lets focus on the second case. Assume that the B-field has a ##B_y## component (y the axis of the pipe). Do we get any info for this ##B_y## component by applying Ampere's law for the loop shown in the second case? NO we don't because ##B_y## is always perpendicular to the ##dl## of the loop (since the loop lies in the x-z plane) so it is ##\vec{B_y}\cdot \vec{dl}=0## because the dot product is always zero as long as the vectors are perpendicular to each other. ##B_y## might be zero or might be any other value but because its dot product will be zero (for this specific loop we choose to apply the law), amperes law hold. That is Ampere's law hold for any choice of ##B_y## for this specific loop.

If we choose the loop of the first case, then we can extract useful info about ##B_y## (because it is not perpendicular everywhere , it is in the same direction as ##dl## in the segments of the loop that are along the y-axis) by applying ampere's law for this loop.

 

[Gourab_chill]

lets focus on the second case. Assume that the B-field has a ##B_y## component (y the axis of the pipe). Do we get any info for this ##B_y## component by applying Ampere's law for the loop shown in the second case? NO we don't because ##B_y## is always perpendicular to the ##dl## of the loop (since the loop lies in the x-z plane) so it is ##\vec{B_y}\cdot \vec{dl}=0## because the dot product is always zero as long as the vectors are perpendicular to each other. ##B_y## might be zero or might be any other value but because its dot product will be zero (for this specific loop we choose to apply the law), amperes law hold. That is Ampere's law hold for any choice of ##B_y## for this specific loop.

If we choose the loop of the first case, then we can extract useful info about ##B_y## (because it is not perpendicular everywhere , it is in the same direction as ##dl## in the segments of the loop that are along the y-axis) by applying ampere's law for this loop.

Thanks :)