Flux of constant magnetic field through lateral surface of cylinder

• lorenz0
In summary: As you move the mug around, the flux of light through the surface will be constant.This makes sense, thank you.
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.

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
Orodruin said:
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?

lorenz0 said:
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
Orodruin said:
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?

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

lorenz0 said:
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##

SammyS
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.

1. What is the flux of a constant magnetic field through the lateral surface of a cylinder?

The flux of a constant magnetic field through the lateral surface of a cylinder is the amount of magnetic field passing through the curved surface of the cylinder. It is measured in units of Weber (Wb) or Tesla (T).

2. How is the flux of a constant magnetic field through the lateral surface of a cylinder calculated?

The flux can be calculated using the formula Φ = B * A * cosθ, where B is the magnetic field strength, A is the area of the curved surface, and θ is the angle between the magnetic field and the normal vector of the surface.

3. What factors affect the flux of a constant magnetic field through the lateral surface of a cylinder?

The flux is affected by the strength of the magnetic field, the area of the curved surface, and the angle between the magnetic field and the normal vector of the surface. It is also affected by the material of the cylinder, as some materials can alter the magnetic field passing through them.

4. How does the direction of the magnetic field affect the flux through the lateral surface of a cylinder?

The direction of the magnetic field is important in determining the flux through the lateral surface of a cylinder. If the magnetic field is perpendicular to the surface, the flux will be at its maximum. As the angle between the magnetic field and the surface increases, the flux decreases.

5. What is the significance of calculating the flux of a constant magnetic field through the lateral surface of a cylinder?

Calculating the flux of a constant magnetic field through the lateral surface of a cylinder is important in understanding the behavior of magnetic fields in different materials and shapes. It is also useful in various applications, such as in the design of motors and generators.

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