Using the fact that we can write the vector cross-product in the form: (A× B)i =? ijk Aj Bk ,
where ?ijk is the Levi-Civita tensor show that:
Del×( fA) = f Del × A - A × Del f ,
where A is a vector function and f a scalar function.
A static magnetic field in a homogeneous, linear, medium with permeability ?0 is given in Cartesian coordinates by:
B(x, y, z) = B0 tanh (x/L) z where L is a constant.
a) Sketch Bz as a function of x/L.
b) What current density, j(x,y,z) produced this magnetic field?
c) Sketch any non-zero components of j as a function of x/L.
Could you please provide as much details in every step as possible; this will help me learn.
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- Thread Starter
- 10-04-2011 09:13
- 10-04-2011 09:44
For question 1, you simply need to apply the suffix notation (the ijk stuff) to the LHS of the equation and the RHS separately. As a hint, in suffix notation, , and .
Also note that I've only considered the i component of curl A (which is del x A). You should do this for both sides of the equation.
For question 2:
Assuming you meant to write , the B field only has a z-component so you simply have to sketch a tanh graph.
As for the current density, I'm not 100% sure on this but because there is no time varying electric field specified you can use the Maxwell equation:
so take the curl of B, divide it by mu0 and that'll give you your current density. The third part should be easy from there.Last edited by trm90; 10-04-2011 at 09:47.
- Thread Starter
- 10-04-2011 11:20
Thanks for the answer, but as i wrote explicitly, i needed the actual steps shown, it would be great help in learning this.