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diff --git a/source/know/concept/electric-field/index.md b/source/know/concept/electric-field/index.md index 38c1ff6..2433edf 100644 --- a/source/know/concept/electric-field/index.md +++ b/source/know/concept/electric-field/index.md @@ -8,43 +8,43 @@ categories: layout: "concept" --- -The **electric field** $\vb{E}$ is a vector field +The **electric field** $$\vb{E}$$ is a vector field that describes electric effects, and is defined as the field that correctly predicts the [Lorentz force](/know/concept/lorentz-force/) -on a particle with electric charge $q$: +on a particle with electric charge $$q$$: $$\begin{aligned} \vb{F} = q \vb{E} \end{aligned}$$ -This definition implies that the direction of $\vb{E}$ +This definition implies that the direction of $$\vb{E}$$ is from positive to negative charges, since opposite charges attracts and like charges repel. -If two opposite point charges with magnitude $q$ +If two opposite point charges with magnitude $$q$$ are observed from far away, they can be treated as a single object called a **dipole**, -which has an **electric dipole moment** $\vb{p}$ defined like so, -where $\vb{d}$ is the vector going from -the negative to the positive charge (opposite direction of $\vb{E}$): +which has an **electric dipole moment** $$\vb{p}$$ defined like so, +where $$\vb{d}$$ is the vector going from +the negative to the positive charge (opposite direction of $$\vb{E}$$): $$\begin{aligned} \vb{p} = q \vb{d} \end{aligned}$$ Alternatively, for consistency with [magnetic fields](/know/concept/magnetic-field/), -$\vb{p}$ can be defined from the aligning torque $\vb{\tau}$ -experienced by the dipole when placed in an $\vb{E}$-field. -In other words, $\vb{p}$ satisfies: +$$\vb{p}$$ can be defined from the aligning torque $$\vb{\tau}$$ +experienced by the dipole when placed in an $$\vb{E}$$-field. +In other words, $$\vb{p}$$ satisfies: $$\begin{aligned} \vb{\tau} = \vb{p} \times \vb{E} \end{aligned}$$ -Where $\vb{p}$ has units of $\mathrm{C m}$. -The **polarization density** $\vb{P}$ is defined from $\vb{p}$, +Where $$\vb{p}$$ has units of $$\mathrm{C m}$$. +The **polarization density** $$\vb{P}$$ is defined from $$\vb{p}$$, and roughly speaking represents the moments per unit volume: $$\begin{aligned} @@ -53,16 +53,16 @@ $$\begin{aligned} \vb{p} = \int_V \vb{P} \dd{V} \end{aligned}$$ -If $\vb{P}$ has the same magnitude and direction throughout the body, -then this becomes $\vb{p} = \vb{P} V$, where $V$ is the volume. -Therefore, $\vb{P}$ has units of $\mathrm{C / m^2}$. +If $$\vb{P}$$ has the same magnitude and direction throughout the body, +then this becomes $$\vb{p} = \vb{P} V$$, where $$V$$ is the volume. +Therefore, $$\vb{P}$$ has units of $$\mathrm{C / m^2}$$. -A nonzero $\vb{P}$ complicates things, -since it contributes to the field and hence modifies $\vb{E}$. +A nonzero $$\vb{P}$$ complicates things, +since it contributes to the field and hence modifies $$\vb{E}$$. We thus define -the "free" **displacement field** $\vb{D}$ -from the "bound" field $\vb{P}$ -and the "net" field $\vb{E}$: +the "free" **displacement field** $$\vb{D}$$ +from the "bound" field $$\vb{P}$$ +and the "net" field $$\vb{E}$$: $$\begin{aligned} \vb{D} \equiv \varepsilon_0 \vb{E} + \vb{P} @@ -70,21 +70,21 @@ $$\begin{aligned} \vb{E} = \frac{1}{\varepsilon_0} (\vb{D} - \vb{P}) \end{aligned}$$ -Where the **electric permittivity of free space** $\varepsilon_0$ is a known constant. +Where the **electric permittivity of free space** $$\varepsilon_0$$ is a known constant. It is important to point out some inconsistencies here: -$\vb{D}$ and $\vb{P}$ contain a factor of $\varepsilon_0$, +$$\vb{D}$$ and $$\vb{P}$$ contain a factor of $$\varepsilon_0$$, and therefore measure **flux density**, -while $\vb{E}$ does not contain $\varepsilon_0$, +while $$\vb{E}$$ does not contain $$\varepsilon_0$$, and thus measures **field intensity**. Note that this convention is the opposite -of the magnetic analogues $\vb{B}$, $\vb{H}$ and $\vb{M}$, -and that $\vb{M}$ has the opposite sign of $\vb{P}$. +of the magnetic analogues $$\vb{B}$$, $$\vb{H}$$ and $$\vb{M}$$, +and that $$\vb{M}$$ has the opposite sign of $$\vb{P}$$. -The polarization $\vb{P}$ is a function of $\vb{E}$. +The polarization $$\vb{P}$$ is a function of $$\vb{E}$$. In addition to the inherent polarity -of the material $\vb{P}_0$ (zero in most cases), +of the material $$\vb{P}_0$$ (zero in most cases), there is a (possibly nonlinear) response -to the applied $\vb{E}$-field: +to the applied $$\vb{E}$$-field: $$\begin{aligned} \vb{P} = @@ -93,10 +93,10 @@ $$\begin{aligned} + \varepsilon_0 \chi_e^{(3)} |\vb{E}|^2 \: \vb{E} + ... \end{aligned}$$ -Where the $\chi_e^{(n)}$ are the **electric susceptibilities** of the medium. -For simplicity, we often assume that only the $n\!=\!1$ term is nonzero, -which is the linear response to $\vb{E}$. -In that case, we define the **absolute permittivity** $\varepsilon$ so that: +Where the $$\chi_e^{(n)}$$ are the **electric susceptibilities** of the medium. +For simplicity, we often assume that only the $$n\!=\!1$$ term is nonzero, +which is the linear response to $$\vb{E}$$. +In that case, we define the **absolute permittivity** $$\varepsilon$$ so that: $$\begin{aligned} \vb{D} @@ -106,15 +106,15 @@ $$\begin{aligned} = \varepsilon \vb{E} \end{aligned}$$ -I.e. $\varepsilon \equiv \varepsilon_r \varepsilon_0$, -where $\varepsilon_r \equiv 1 + \chi_e^{(1)}$ is +I.e. $$\varepsilon \equiv \varepsilon_r \varepsilon_0$$, +where $$\varepsilon_r \equiv 1 + \chi_e^{(1)}$$ is the [**dielectric function**](/know/concept/dielectric-function/) or **relative permittivity**, whose calculation is of great interest in physics. -In reality, a material cannot respond instantly to $\vb{E}$, -meaning that $\chi_e^{(1)}$ is a function of time, -and that $\vb{P}$ is the convolution of $\chi_e^{(1)}(t)$ and $\vb{E}(t)$: +In reality, a material cannot respond instantly to $$\vb{E}$$, +meaning that $$\chi_e^{(1)}$$ is a function of time, +and that $$\vb{P}$$ is the convolution of $$\chi_e^{(1)}(t)$$ and $$\vb{E}(t)$$: $$\begin{aligned} \vb{P}(t) @@ -122,6 +122,6 @@ $$\begin{aligned} = \varepsilon_0 \int_{-\infty}^\infty \chi_e^{(1)}(t - \tau) \: \vb{E}(\tau) \:d\tau \end{aligned}$$ -Note that this definition requires $\chi_e^{(1)}(t) = 0$ for $t < 0$ +Note that this definition requires $$\chi_e^{(1)}(t) = 0$$ for $$t < 0$$ in order to ensure causality, which leads to the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations/). |