Consider a parallel plate capacitor of two plates separated by a distance $d$. The plates are connected to a power supply with potential difference $\Delta V$, until the plates carry charges $+Q$ and $-Q$ respectively. The power supply is then disconnected leaving the potential difference between the plates.
🙉 process: if we take a Gaussian surface around each of the plates we can determine the electric field under the assumption that there is a vacuum between the plates
Gauss’ law gives
$$ \int\vec E\cdot\text d\vec S=E_z A=\frac{Q}{\epsilon_0} $$
where $A$ is the surface area of the plates and we have aligned the system along the $z$-axis (the plate is in $x$-$y$ plane)
A constant electric field
$$ E_z=\frac{Q}{A \epsilon_0} $$
The potential difference between the top and bottom plates is
$$ \Delta V=E_z d=\frac{Qd}{A\epsilon_0} $$
We define capacitance to be the ratio between the charge and potential difference
$$ C=\frac{Q}{\Delta V}=\frac{\epsilon_0 A}{d} $$
whose unit is the Farad [$\mathrm{F}$]. This is a property of the system which is only dependent on the geometry of the system, in this case only on $A$ and $d$.
The energy of a parallel plate capacitor
$$ U=\frac 12 \epsilon_0 \int |\vec E|^2 \text dV=\frac 12 \epsilon_) \left ( \frac{\Delta V}{d} \right )^2 Ad=\frac 12 C(\Delta V)^2 $$
If we now add a material between the plates the measured voltage is observed to drop, meaning that the Capacitance increases.
<aside> 🥪 Relative permittivity (dielectric constant):
$$ \epsilon_r=\frac{C}{C_\text{vacuum}} $$
$C$ is the measured value of the capacitance with some material separating the plates
$C_\text{vacuum}$ is the capacitance for the same geometry, that is the same physical separation and shape of the parallel plates, but separated by a vacuum.
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<aside> 🥪 Dielectrics: electrical insulators, meaning that the materials have low conductivity, and the electrons are bound into atoms and molecules. Hence, there are few free electrons. When an electric field is applied to a dielectric the intrinsic dipoles within the material can become aligned, while atoms/molecules with no intrinsic dipole moment become polarized creating dipole.
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<aside> 💿 Polarization: of a material object occurs when the constituents of the substance align in some preferred direction associated with an electric field.
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Consider the action of a single electric dipole in an external electric field $\vec E_\text{ext}$.
🧠 Remember: the dipole moment is $\vec p=q\vec d$
The work done by the dipole is the energy change due to the dipole
$$ U_\text{ext}=-qE_\text{ext}d\cos\theta $$
where $\theta$ is the angle between the dipole and the external field.
The torque on the dipole is
$$ \tau_\text{ext}=Fd\sin\theta=dq E_\text{ext}\sin\theta $$
🗒️ Note: if $\theta=0$ the energy due to the external field is minimized and the torque is zero Therefore the dipoles will attempt to rotate to align with the applied field to minimize the energy and eliminate torque
For a general dipole, the energy due to and torque on a dipole in an applied electric field $\vec E_\text{ext}$ are calculated via
$$ U_\text{ext}=-\vec p\cdot \vec E_\text{ext} \qquad \vec \tau_\text{ext}=\vec p \times \vec E_\text{ext} $$
the lowest energy state is achieved by setting $\vec p \propto \vec E_\text{ext}$ which corresponds to a state with zero torque
💎 Conclusion: when an electric field is applied to a material we get