Introduction to Electrodynamics · Problem 10.10

Potentials and Fields — David J. Griffiths

Physics · retarded-potentials-lorenz-gauge

Problem

Confirm that the retarded potentials satisfy the Lorenz gauge condition. [Hint: First show that ∇⋅(Ji)=1i(∇⋅J)+1i(∇′⋅J)−∇′⋅(Ji),\nabla \cdot \left(\frac{\mathbf{J}}{i}\right) = \frac{1}{i}(\nabla \cdot \mathbf{J}) + \frac{1}{i}(\nabla' \cdot \mathbf{J}) - \nabla' \cdot \left(\frac{\mathbf{J}}{i}\right), where ∇\nabla denotes derivatives with respect to r\mathbf{r}, and ∇′\nabla' denotes derivatives with respect to r′\mathbf{r}'. Next, noting that J(r′,t−i/c)\mathbf{J}(\mathbf{r}', t - i/c) depends on r\mathbf{r} both explicitly and through ii, whereas it depends on r′\mathbf{r}' only through ii, confirm that ∇⋅J=−1cJ˙⋅(∇i),∇′⋅J=−ρ˙−1cJ˙⋅(∇′i).\nabla \cdot \mathbf{J} = -\frac{1}{c}\dot{\mathbf{J}} \cdot (\nabla i), \quad \nabla' \cdot \mathbf{J} = -\dot{\rho} - \frac{1}{c}\dot{\mathbf{J}} \cdot (\nabla' i). Use this to calculate the divergence of A\mathbf{A} (Eq. 10.26).]

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