15D (Electromagnetism)

For

X^{μ} = (c t, x, y, z)

t^{'} = γ (t - \frac{vx}{c ^{2}}), x^{'} = γ (x - v t), y^{'} = y, z^{'} = z .

b

F^{'} = Λ F Λ^{T}

and reading off its components gives

E_{1}^{'} = E_{1}, B_{1}^{'} = B_{1}, E_{2}^{'} = γ (E_{2} - v B_{3}), B_{2}^{'} = γ (B_{2} + \frac{v E _{3}}{c ^{2}}), E_{3}^{'} = γ (E_{3} + v B_{2}), B_{3}^{'} = γ (B_{3} - \frac{v E _{2}}{c ^{2}}) .

These are the parallel and transverse field transformation laws for a boost in the

x

direction.

Take the wire along the

x

-axis and let

r

E = \frac{λ}{2 π ε _{0} r} e_{r}, B = 0.

For an observer boosted parallel to the wire,

E^{'} = γ \frac{λ}{2 π ε _{0} r} e_{r}, B^{'} = - \frac{γ}{c ^{2}} v \times E .

The moving observer sees length contraction and hence line density

λ^{'} = γλ

, together with current

I^{'} = - γλ v

. The transformed magnetic field is exactly

B^{'} = \frac{μ _{0} ∣ I ^{'} ∣}{2 π r},

with the direction prescribed by the current. This is Ampère's law for the current seen in the moving frame.

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