Flux of constant magnetic field through lateral surface of cylinder

[lorenz0]

Homework Statement

In a region of space there is a constant magnetic field $\vec{B}=B(3,2,1)$, where $B$ is constant. What is the flux of the magnetic field through the lateral surface of a cylinder present in that region of space?

Relevant Equations

$\phi_{S\text{ closed}}(\vec{B})=\int_{S\text{ closed}}\vec{B}\cdot d\vec{S}=0$

If the question had been asking about the flux through the whole surface of the cylinder I would have said that the flux is 0, but since it is asking only about the lateral surfaces I am wondering how one could calculate such a flux not knowing how the cylinder is oriented in space. One could for example take a cylinder whose axis lies on the line spanned by the vector $(3,2,1)$ and say that the flux through the lateral surface of the cylinder is 0. So, the answer in this case depends on the orientation of the cylinder in space, am I right?
Or is there a way to find out the flux even without knowing anything more specific about the cylinder?
Thanks.

 

[Orodruin]

You have (correctly) deduced that the flux through the full cylinder surface is zero. What can you say about the flux through the ends of the cylinder?

 

[lorenz0]

You have (correctly) deduced that the flux through the full cylinder surface is zero. What can you say about the flux through the ends of the cylinder?

That the flux through one is the opposite of the flux through the other in the case that I have described where the axis of the cylinder lies on the line spanned by the vector $(3,2,1)$ but can I say anything more without knowing something more about the orientation of the cylinder in space?

 

[Orodruin]

That the flux through one is the opposite of the flux through the other in the case that I have described where the axis of the cylinder lies on the line spanned by the vector $(3,2,1)$ but can I say anything more without knowing something more about the orientation of the cylinder in space?

Why only if the cylinder lies on that axis?

 

[lorenz0]

Why only if the cylinder lies on that axis?

Ah, thinking more about it, could I say that since the $\vec{B}$-field is constant across space and since the cylinder is symmetric I would have that the flux through the lateral surface of the cylinder is $0$, because when "entering" the cylinder it has a plus sign and when "exiting" through the other end it has a - sign and the same magnitude. Does this make sense?

 

[Orodruin]

I don't think that is very rigorous. Can you make your argument more formalised?

 

[kuruman]

Ah, thinking more about it, could I say that since the $\vec{B}$-field is constant across space and since the cylinder is symmetric I would have that the flux through the lateral surface of the cylinder is $0$, because when "entering" the cylinder it has a plus sign and when "exiting" through the other end it has a - sign and the same magnitude. Does this make sense?

You might consider revisiting @Orodruin's question in post #2. You have established that the flux through the entire cylinder is zero regardless of the cylinder's orientation. If you can show that the flux through the two flat ends is also zero regardless of the cylinder's orientation, then $\dots$

 

[bob012345]

It also might help you to visualize this by taking a cylinder like object such as a coffee mug and holding it up against a light.