局域化的時變电荷 和电流密度 在真空中是电磁波 的源。在有源的情形下，麦克斯韦方程组 可以写成一个非齐次的电磁波方程 （英文：Inhomogeneous electromagnetic wave equation 偏微分方程 变为非齐次。
真空中的麦克斯韦方程组在含有电荷 ρ {\displaystyle \rho } J {\displaystyle \mathbf {J} } 矢势 和标势 表示为
∇ 2 φ + ∂ ∂ t ( ∇ ⋅ A ) = − ρ ε 0 {\displaystyle \nabla ^{2}\varphi +{{\partial } \over \partial t}\left(\nabla \cdot \mathbf {A} \right)=-{\rho \over \varepsilon _{0}}} ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 − ∇ ( 1 c 2 ∂ φ ∂ t + ∇ ⋅ A ) = − μ 0 J {\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}-\nabla \left({1 \over c^{2}}{{\partial \varphi } \over {\partial t}}+\nabla \cdot \mathbf {A} \right)=-\mu _{0}\mathbf {J} } 此时电场和磁场分别为
E = − ∇ φ − ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla \varphi -{\partial \mathbf {A} \over \partial t}} 以及
B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } 如果加上洛伦茨规范条件
1 c 2 ∂ φ ∂ t + ∇ ⋅ A = 0 {\displaystyle {1 \over c^{2}}{{\partial \varphi } \over {\partial t}}+\nabla \cdot \mathbf {A} =0} 则非齐次的波动方程 为
∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = − ρ ε 0 {\displaystyle \nabla ^{2}\varphi -{1 \over c^{2}}{\partial ^{2}\varphi \over \partial t^{2}}=-{\rho \over \varepsilon _{0}}} ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 = − μ 0 J {\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu _{0}\mathbf {J} } 在厘米-克-秒制 下，方程的形式为
∇ 2 φ − 1 c 2 ∂ 2 φ ∂ t 2 = − 4 π ρ {\displaystyle \nabla ^{2}\varphi -{1 \over c^{2}}{\partial ^{2}\varphi \over \partial t^{2}}=-{4\pi \rho }} ∇ 2 A − 1 c 2 ∂ 2 A ∂ t 2 = − 4 π c J {\displaystyle \nabla ^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-{4\pi \over c}\mathbf {J} } 电场和磁场的形式为
E = − ∇ φ − 1 c ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla \varphi -{1 \over c}{\partial \mathbf {A} \over \partial t}} B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } 洛伦茨规范条件为
1 c ∂ φ ∂ t + ∇ ⋅ A = 0 {\displaystyle {1 \over c}{{\partial \varphi } \over {\partial t}}+\nabla \cdot \mathbf {A} =0} 如果采取有时在高维相对论场合计算中使用的洛伦兹-赫维赛德单位制 ，电荷和电流密度需要从厘米-克-秒制变换为
ρ → ρ 4 π {\displaystyle \rho \rightarrow {\rho \over {4\pi }}} J → 1 4 π J {\displaystyle \mathbf {J} \rightarrow {1 \over {4\pi }}\mathbf {J} } 在狭义相对论 中，麦克斯韦方程组可以写成协变 的形式：
◻ A μ = d e f ∂ β ∂ β A μ = d e f A μ , β β = − μ 0 J μ {\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-\mu _{0}J^{\mu }} ◻ A μ = d e f ∂ β ∂ β A μ = d e f A μ , β β = − 4 π c J μ {\displaystyle \Box A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ \partial _{\beta }\partial ^{\beta }A^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {A^{\mu ,\beta }}_{\beta }=-{\frac {4\pi }{c}}J^{\mu }} （厘米-克-秒制）
其中 J μ {\displaystyle J^{\mu }\,} 四维电流密度 ：
J μ = ( c ρ , J ) {\displaystyle J^{\mu }=\left(c\rho ,\mathbf {J} \right)} ∂ ∂ x a = d e f ∂ a = d e f , a = d e f ( ∂ / ∂ c t , ∇ ) {\displaystyle {\partial \over {\partial x^{a}}}\ {\stackrel {\mathrm {def} }{=}}\ \partial _{a}\ {\stackrel {\mathrm {def} }{=}}\ {}_{,a}\ {\stackrel {\mathrm {def} }{=}}\ (\partial /\partial ct,\nabla )} 是四维梯度 ，而电磁四维势 为
A μ = ( φ , A c ) {\displaystyle A^{\mu }=(\varphi ,\mathbf {A} c)} A μ = ( φ , A ) {\displaystyle A^{\mu }=(\varphi ,\mathbf {A} )} 洛伦茨规范为
∂ μ A μ = 0 {\displaystyle \partial _{\mu }A^{\mu }=0} 这里
◻ = ∂ β ∂ β = ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 {\displaystyle \Box =\partial _{\beta }\partial ^{\beta }=\nabla ^{2}-{1 \over c^{2}}{\frac {\partial ^{2}}{\partial t^{2}}}} 达朗贝尔算符 。电磁波方程在弯曲时空中需要做两处修正，分别是偏导数被替换为协变导数 ，以及增加了一项有关时空曲率的项。在国际单位制下
− A α ; β β + R α β A β = μ 0 J α {\displaystyle -{A^{\alpha ;\beta }}_{\beta }+{R^{\alpha }}_{\beta }A^{\beta }=\mu _{0}J^{\alpha }} 其中
R α β {\displaystyle {R^{\alpha }}_{\beta }} 是里奇曲率张量 。 这里分号表示对角标求协变导数。对于厘米-克-秒制下的方程，需要用 4 π / c {\displaystyle 4\pi /c} 真空磁导率 。
这里假设洛伦茨规范在弯曲时空中的推广为
A μ ; μ = 0 {\displaystyle {A^{\mu }}_{;\mu }=0} 在波源周围没有边界条件 的情形下，非齐次波方程在厘米-克-秒制下的解为
φ ( r , t ) = ∫ δ ( t ′ + | r − r ′ | c − t ) | r − r ′ | ρ ( r ′ , t ′ ) d 3 r ′ d t ′ {\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'} 以及
A ( r , t ) = ∫ δ ( t ′ + | r − r ′ | c − t ) | r − r ′ | J ( r ′ , t ′ ) c d 3 r ′ d t ′ {\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'} 其中
δ ( t ′ + | r − r ′ | c − t ) {\displaystyle {\delta \left(t'+{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)}} 是狄拉克δ函数 。
对于国际单位制，
ρ → ρ 4 π ε 0 {\displaystyle \rho \rightarrow {\rho \over {4\pi \varepsilon _{0}}}} J → μ 0 4 π J {\displaystyle \mathbf {J} \rightarrow {\mu _{0} \over {4\pi }}\mathbf {J} } 对于洛伦兹-赫维赛德单位制，
ρ → ρ 4 π {\displaystyle \rho \rightarrow {\rho \over {4\pi }}} J → 1 4 π J {\displaystyle \mathbf {J} \rightarrow {1 \over {4\pi }}\mathbf {J} } 这些解被称作推迟解，它们表示的是一族由波源向外发出的并从现在向未来传播的球面电磁波的线性叠加 。
此外还有所谓超前解，表示为
φ ( r , t ) = ∫ δ ( t ′ − | r − r ′ | c − t ) | r − r ′ | ρ ( r ′ , t ′ ) d 3 r ′ d t ′ {\displaystyle \varphi (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}\rho (\mathbf {r} ',t')d^{3}r'dt'} 以及
A ( r , t ) = ∫ δ ( t ′ − | r − r ′ | c − t ) | r − r ′ | J ( r ′ , t ′ ) c d 3 r ′ d t ′ {\displaystyle \mathbf {A} (\mathbf {r} ,t)=\int {{\delta \left(t'-{{\left|\mathbf {r} -\mathbf {r} '\right|} \over c}-t\right)} \over {\left|\mathbf {r} -\mathbf {r} '\right|}}{\mathbf {J} (\mathbf {r} ',t') \over c}d^{3}r'dt'} 它们表示的是一族由波源向外发出的并从未来向现在传播的球面电磁波的线性叠加。
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