Skip to main content
deleted 11 characters in body
Source Link
user36790
user36790

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F ~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ U &\equiv \textrm{Potential-energy}, \\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ U &\equiv \textrm{Potential-energy}, \\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F ~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ U &\equiv \textrm{Potential-energy}, \\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

added 42 characters in body
Source Link
user36790
user36790

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ U &\equiv \textrm{Potential-energy}, \\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ U &\equiv \textrm{Potential-energy}, \\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

added 12 characters in body
Source Link
user36790
user36790

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathrm 2, t')}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t' &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathrm 2, t')}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t' &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

$$\begin{array}{|c|c|} \hline\textrm{True in Statics} &\textrm{True in General}\\ \hline \mathbf F = \mathbf E~= \dfrac1{4\pi\varepsilon_o}~ \dfrac{q_1q_2}{r^2}~\mathbf{\hat r} & \mathbf F= q(\mathbf E+ \mathbf v\times \mathbf B)\\ \hline \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} & \nabla \cdot \mathbf E = \dfrac{\rho}{\varepsilon_0} \\ \hline \nabla\times \mathbf E= \mathbf 0& \nabla \times \mathbf E = ~-\partial_t\mathbf B\\ \hline \mathbf E= -\nabla\varphi & \mathbf E= -\nabla \varphi - \partial_t \mathbf A\\ \hline \nabla^2\varphi = -\dfrac{\rho}{\varepsilon_0} & \nabla^2\varphi -\dfrac1 {c^2}\partial^2_t\varphi = -\dfrac\rho{\varepsilon_0}\\ \hline \varphi(\mathbf 1)= \dfrac1{4\pi\varepsilon_0}\displaystyle\int\dfrac{\rho(\mathbf 2)}{r_{12}}~\mathrm dV_2 & \varphi(\mathbf 1, t)= \dfrac{1}{4\pi\varepsilon_0}\displaystyle \int \dfrac{\rho(\mathbf 2, t^\prime)}{r_{12}}~\mathrm dV_2\\\hline U= \dfrac12 \left(\displaystyle\int \rho\varphi~\mathrm dV \right) & U= \dfrac{\varepsilon_0}2 \left(\displaystyle\int \mathbf E\cdot \mathbf E~\mathrm dV + {c^2}\displaystyle\int \mathbf B\cdot \mathbf B~\mathrm dV\right)\\ \hline\end{array}$$

where

\begin{align}\mathbf F& \equiv \textrm{Total force},\\ \mathbf E& \equiv \textrm{Electric field},\\ \mathbf B &\equiv \textrm{Magnetic field},\\ \nabla &\equiv \textrm{del operator},\\ \rho &\equiv\textrm{Charge-density},\\ \varphi &\equiv \textrm{Scalar-potential},\\ \mathbf A &\equiv \textrm{Vector-potential},\\ \partial_t &\equiv \dfrac{\partial}{\partial t},\\ \nabla^2 &\equiv \textrm{Lapalacian operator},\\ \partial^2_t &\equiv \dfrac{\partial^2}{\partial t^2},\\ t^\prime &\equiv \textrm{retarded-time} = t-\dfrac{r_{12}}c\;. \end{align}

added 1 character in body
Source Link
user36790
user36790
Loading
added 503 characters in body
Source Link
user36790
user36790
Loading
added 503 characters in body
Source Link
user36790
user36790
Loading
Source Link
user36790
user36790
Loading