In the previous problem, you derived the Lorentz transformations of the B field starting with the assumption that the scalar and vector potentials are components of a four vector \(A^{\mu}=(\phi, \mathbf{A})\). Using a similar approach, derive the Lorentz transformation of the electric field \(\mathbf{E}\); show that you get 

and

Note that this is a more involved computation than in the previous problem.

Data from previous problem

We discovered in the text that the scalar and vector potentials are components of a four vector \(A^{\mu}=(\phi, \mathbf{A})\). In this problem, we will take as given the existence of this four-vector potential \(A^{\mu}\) and, using the known Lorentz transformation of a four-vector and the relations of \(A^{\mu}\) to \(\mathbf{E}\) and \(\mathbf{B}\), we want to derive the Lorentz transformations of \(\mathbf{E}\) and \(\mathbf{B}\). Consider two inertial frames \(\mathcal{O}\) and \(\mathcal{O}^{\prime}\) where \(\mathcal{O}^{\prime}\) is moving with velocity v relative to \(\mathcal{O}\). We split all three-vectors in components parallel and perpendicular to the direction of the Lorentz boost, \(\mathbf{v}\) : for example, we have \(\mathbf{E}=\mathbf{E}_{\|}+\mathbf{E}_{\perp}\). Note that the gradient vector can also be decomposed as \(abla=abla_{\|}+abla_{\perp}\).

(a) First show that \(\mathbf{B}_{\|}^{\prime}=\mathbf{B}_{\|}\).

(b) Show next that \(\boldsymbol{abla}_{\|}^{\prime}=\gamma\left(\boldsymbol{abla}_{\|}+\left(\mathbf{v} / c^{2}\right)(\partial / \partial t)\right)\).

(c) Finally, show that

as in the previous problem.

Transcribed Image Text:

E = E||