Question

Show that the substitution of the Lagrangian $$ \mathcal{L}=-{ }_4^1 F_{\mu v} F^{\mu \nu}-j^\mu A_\mu $$ into the Euler-Lagrange equation for $A_\mu$ gives the Maxwell equations, $(6.57)$, $$ \partial_\mu F^{\mu \nu}=j^\nu, $$ where $F^{\mu v} \equiv \partial^\mu A^v-\partial^v A^\mu$. Hence, show that the current is conserved, that is, $\partial_v j^v=0$.

   Show that the substitution of the Lagrangian
$$
\mathcal{L}=-{ }_4^1 F_{\mu v} F^{\mu \nu}-j^\mu A_\mu
$$
into the Euler-Lagrange equation for $A_\mu$ gives the Maxwell equations, $(6.57)$,
$$
\partial_\mu F^{\mu \nu}=j^\nu,
$$
where $F^{\mu v} \equiv \partial^\mu A^v-\partial^v A^\mu$. Hence, show that the current is conserved, that is, $\partial_v j^v=0$.

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Quarks and leptons: introductory course in modern particle physics

Quarks and leptons: introductory course in modern particle physics

Francis Halzen, Alan… 1st Edition

Chapter 14, Problem 3

Instant Answer

Step 1

Step 1: Start by writing down the Lagrangian for the electromagnetic field interacting with a current density \( j^\mu \): \[ \mathcal{L} = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} - j^\mu A_\mu, \] where \( F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \) is Show more…

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Show that the substitution of the Lagrangian $$ \mathcal{L}=-{ }_4^1 F_{\mu v} F^{\mu \nu}-j^\mu A_\mu $$ into the Euler-Lagrange equation for $A_\mu$ gives the Maxwell equations, $(6.57)$, $$ \partial_\mu F^{\mu \nu}=j^\nu, $$ where $F^{\mu v} \equiv \partial^\mu A^v-\partial^v A^\mu$. Hence, show that the current is conserved, that is, $\partial_v j^v=0$.

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