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Tensor and Megnetism Questions HELP!!! watch

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    (((Q1)))

    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.


    (((Q2)))

    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.


    Thanks
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    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, (\nabla \times A)_i = \epsilon_{ijk} \dfrac{\partial A_j}{\partial x_k}, and (\nabla f)_i = \dfrac{\partial f}{\partial x_i}.

    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 \vec{B} (x,y,z) = B_0 \tanh(x/L) \vec{z}, 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:

    \nabla \times \vec{B} = \mu_0 \vec{J}

    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.
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    Thanks for the answer, but as i wrote explicitly, i needed the actual steps shown, it would be great help in learning this.
 
 
 
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