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thatguy14

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## Homework Statement

We will see (in Chap. 5) that the magnetic field can be derived from a vector potential function as

follows:

B = ∇×A

Show that, in the special case of a uniform magnetic field B[itex]_{0}[/itex] , one possible

vector potential function is A = [itex]\frac{1}{2}[/itex]B[itex]_{0}[/itex]×

**r**

MUST USE TENSOR NOTATIONm also B0 is constant (uniform magnetic field)

## Homework Equations

ε[itex]_{ijk}[/itex]ε[itex]_{klm}[/itex] = δ[itex]_{il}[/itex]δ[itex]_{jm}[/itex] - δ[itex]_{im}[/itex]δ[itex]_{jl}[/itex]

## The Attempt at a Solution

I have tried a bunch of different things but I am missing something near the end.

Here is what I have

B = (∇×A)[itex]_{i}[/itex]

B = ε[itex]_{ijk}[/itex]∂[itex]_{j}[/itex]A[itex]_{k}[/itex]

B = ε[itex]_{ijk}[/itex]∂[itex]_{j}[/itex] ([itex]\frac{1}{2}[/itex]ε[itex]_{klm}[/itex]B[itex]_{0l}[/itex]r[itex]_{m}[/itex])

B = [itex]\frac{1}{2}[/itex]ε[itex]_{ijk}[/itex]ε[itex]_{klm}[/itex]∂[itex]_{j}[/itex]B[itex]_{0l}[/itex]r[itex]_{m}[/itex])

where ε[itex]_{ijk}[/itex]ε[itex]_{klm}[/itex] = δ[itex]_{il}[/itex]δ[itex]_{jm}[/itex] - δ[itex]_{im}[/itex]δ[itex]_{jl}[/itex]

So B = [itex]\frac{1}{2}[/itex][δ[itex]_{il}[/itex]δ[itex]_{jm}[/itex] - δ[itex]_{im}[/itex]δ[itex]_{jl}[/itex]]∂[itex]_{j}[/itex]B[itex]_{0l}[/itex]r[itex]_{m}[/itex]

Changing indicies gives (noting that the derivative of constant = 0 and using the product rule)

B = [itex]\frac{1}{2}[/itex][B[itex]_{0i}[/itex]∂[itex]_{m}[/itex]r[itex]_{m}[/itex] - B[itex]_{0l}[/itex]∂[itex]_{l}[/itex]r[itex]_{i}[/itex]]

And that's where I am stuck. What comes next? I am assuming that with the last term there, l and i have to be equal (because if they aren't then it equals 0) and I think I have to introduct the krockner delta somewhere but I am unsure. Any help would be greatly appreciated.

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