Magnetic field strength and magnetic flux

[Jimmy87]

Hi, please could someone help with my confusion over these two qauntities. In class we recently learned that magnetic field strength (B) is the number of flux lines (measured in Webers) per square metre. The magnetic flux on the other hand is the total number of flux lines (measured in Webers), hence the equation for magnetic flux is:

Magnetic flux = B A

What I don't quite get is why you need to take the dot product with the area vector. Flux is associated with something that penetrates a given surface. But magnetic field strength is webers/square metre therefore it is the number of flux lines per square metre. So don't flux lines by definition already penetrate a surface, hence flux lines per square metre already referes to a magnetic field that is penetrating a surface?

 

[Orodruin]

The field strength is the flux through a unit area orthogonal to the field itself. If the area is parallel to the flux, no flux lines will cross it.

Try the following (in 2D, it is difficult to draw in 3D):
- Draw evenly spaced flux lines on a paper.
- Draw a line of a certain length orthogonal to the flux lines, count the number of lines crossing the line.
- Now draw a line of the same length parallel to the flux lines, how many flux lines cross this line?
- Draw yet another line of the same length, this time in a 45 degree angle to the flux lines. How many flux lines cross this line?

 

[Jimmy87]

The field strength is the flux through a unit area orthogonal to the field itself. If the area is parallel to the flux, no flux lines will cross it.

Try the following (in 2D, it is difficult to draw in 3D):
- Draw evenly spaced flux lines on a paper.
- Draw a line of a certain length orthogonal to the flux lines, count the number of lines crossing the line.
- Now draw a line of the same length parallel to the flux lines, how many flux lines cross this line?
- Draw yet another line of the same length, this time in a 45 degree angle to the flux lines. How many flux lines cross this line?

Thanks for your reply. Yeh that makes sense but the problem I have is that flux is defined as field lines that enter a surface at 90 degrees. Magnetic field strength (flux density) is the FLUX per unit area. Therefore, the field lines will already be at 90 degrees because you are talking about FLUX per unit area. Therefore, it makes not sense to me to take the quantity B which is orthogonal to the surface because it will already be orthogonal as its flux. It would make more sense to me if magnetic field strength is the number of field lines per unit area. Then flux would be the proportion of these field lines that are orthogonal to the area (or parallel to the area vector).

 

[Orodruin]

Thanks for your reply. Yeh that makes sense but the problem I have is that flux is defined as field lines that enter a surface at 90 degrees.

Well, this is more or less the point. The quantity $\vec B$ is a vector, when you define it as above you are defining the component of that vector in the normal direction of the surface. If you know $\vec B$ and want to compute the flux through a given area, you can decompose $\vec B$ in a component parallel to the area and another which is orthogonal to the area. The component parallel to the area will not contribute with any flux through the area, while the orthogonal component is just what you defined above. It is this decomposition that gives you the scalar product between the unit normal to the surface and the $\vec B$ field.

 

[Jimmy87]

Well, this is more or less the point. The quantity $\vec B$ is a vector, when you define it as above you are defining the component of that vector in the normal direction of the surface. If you know $\vec B$ and want to compute the flux through a given area, you can decompose $\vec B$ in a component parallel to the area and another which is orthogonal to the area. The component parallel to the area will not contribute with any flux through the area, while the orthogonal component is just what you defined above. It is this decomposition that gives you the scalar product between the unit normal to the surface and the $\vec B$ field.

Thanks. Yeh I kind of get your point. The fact that magnetic field strength is the flux per unit area seems like there is no need to resolve it into a component parallel and orthogonal to the area because flux itself by definition is already orthogonal.