Electrostatics and Capacitance

Name	Formula	Related Formulae	Variables/Constants
Coulomb’s law	$F=\frac{1}{4\pi {ϵ}_0}\frac{q_1{q}_2}{r^2}$	$k_e=\frac{1}{4\pi {ϵ}_0}\approx 8.99\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$	$F:forcebetweenpointcharges$ $q_1,q_2:charges$ $r:separation$ ${ϵ}_0:permittivityoffreespace$ $k_e:Coulombconstant$
Electric field	$\vec{E}=\frac{\vec{F}}{q}$	$E=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r^2}(\text{point charge}),E=\frac{\sigma }{2{ϵ}_0}(\text{infinite sheet})$	$\vec{E}:electricfield$ $\vec{F}:forceonatestchargeq$ $Q:sourcecharge$ $r:distance$ $\sigma :surfacechargedensity$
Gauss’s law	$\oint \vec{E}\cdot d\vec{A}=\frac{Q_{enc}}{{ϵ}_0}$	$\nabla \cdot \vec{D}={\rho }_v,\vec{D}=ϵ\vec{E}$	$d\vec{A}:outwardareaelement$ $Q_{enc}:enclosedcharge$ $\vec{D}:electricdisplacement$ ${\rho }_v:volumechargedensity$ $ϵ:permittivity$
Electric potential	$V_B-V_A=-{\int }_A^B\vec{E}\cdot d\vec{l}$	$\vec{E}=-\nabla V,V=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}(\text{point charge})$	$V:electricpotential$ $\vec{E}:electricfield$ $d\vec{l}:pathelement$ $Q:charge$ $r:distance$
Potential energy and the electron-volt	$\Delta U=q\Delta V$	$1\mathrm{eV}=1.602\times 10^{-19}\mathrm{J}$	$\Delta U:changeinpotentialenergy$ $q:charge$ $\Delta V:potentialdifference$ $\begin{array}{r}eV:energygainedbyoneelectroncharge\\ acrossonevolt\end{array}$
Poisson’s equation	${\nabla }^{2}V=-\frac{{\rho }_v}{ϵ}$	${\nabla }^{2}V=0(\text{Laplace, charge-free region})$	$V:potential$ ${\rho }_v:volumechargedensity$ $ϵ:permittivity$
Capacitance	$C=\frac{Q}{V}$	$i=C\frac{dv}{dt}$	$C:capacitance$ $Q:chargeoneachplate$ $V:voltagebetweenplates$ $i:current$ $v:voltage$ $t:time$
Parallel-plate capacitor	$C=\frac{{ϵ}_r{ϵ}_{0}A}{d}$	$C_{ox}=\frac{{ϵ}_{ox}}{t_{ox}}(\text{per unit area})$	$\begin{array}{r}{ϵ}_r:relativepermittivity(dielectric\\ constant)\end{array}$ ${ϵ}_0:permittivityoffreespace$ $A:platearea$ $d:plateseparation$ $C_{ox}:oxidecapacitanceperunitarea$ ${ϵ}_{ox}:oxidepermittivity$ $t_{ox}:oxidethickness$
Series and parallel capacitors	$C_{eq}=C_1+C_2+\dots (\text{parallel})$	$\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\dots (\text{series})$	$C_{eq}:equivalentcapacitance$ $C_1,C_2:individualcapacitances$
Energy stored in a capacitor	$U=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}$	$u_E=\frac{1}{2}ϵE^2$	$U:storedenergy$ $C:capacitance$ $V:voltage$ $Q:charge$ $u_E:energydensity$ $ϵ:permittivity$ $E:electricfield$

Important Questions

Ques: What is the fastest way to remember Q = CV?

Read capacitance as charge per volt. A capacitor is like a bucket whose size C says how much charge it holds for each volt applied, so $Q=CV$, and since current is the rate of flow of charge, $i=Cdv/dt$. The unit confirms it: a farad is a coulomb per volt. The parallel-plate formula $C={ϵ}_r{ϵ}_{0}A/d$ shows what makes the bucket bigger: larger plates, a thinner gap and a dielectric with higher permittivity. A farad is enormous: 1 pF needs plates of 1 mm^2 only 8.85 $\mu$m apart in vacuum (computed). The same pattern appears for inductors, where flux linkage per ampere gives $N\Phi =LI$, on the Magnetism, Flux and Inductance page. Reference: D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014); W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. (McGraw-Hill, 2012).

Ques: Is the field of a charged sheet sigma over epsilon0 or sigma over two epsilon0?

Both are correct, for different situations. An isolated infinite sheet of charge sends half of its field lines to each side, so applying Gauss’s law to a small box straddling it gives $2EA=\sigma A/{ϵ}_0$, or $E=\sigma /(2{ϵ}_0)$. At the surface of a conductor the field inside is zero, so all the flux leaves through one face and $E=\sigma /{ϵ}_0$. A parallel-plate capacitor is two sheets with opposite charge; their fields add between the plates to $\sigma /{ϵ}_0$ and cancel outside, which is how $C={ϵ}_{0}A/d$ is derived. Mixing up the two cases is a common factor-of-two error in electrostatics problems. The conductor result is also why charge on a metal gate sits entirely at its surface. Reference: D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013); W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. (McGraw-Hill, 2012).

Ques: Why does a dielectric increase capacitance?

Because it polarizes and partly cancels the field. The bound charges of the dielectric create a field opposing the applied one, so for a given charge on the plates the voltage drops by the factor ${ϵ}_r$, and the charge stored per volt rises by the same factor, $C={ϵ}_r{ϵ}_{0}A/d$. In a MOSFET the gate oxide is such a capacitor: silicon dioxide with ${ϵ}_r=3.9$ and a thickness of 2 nm gives $C_{ox}={ϵ}_{ox}/t_{ox}$ of about $1.7\times 10^{-2}$ F/m^2, or 17 fF/$\mu$m^2 (computed). High-permittivity dielectrics such as hafnium oxide give the same capacitance with a thicker film, which reduces tunnelling leakage. This oxide capacitance sets the inversion charge in the MOS Capacitor page. Reference: B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed. (Pearson, 2015); S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); D. J. Griffiths, Introduction to Electrodynamics, 4th ed. (Pearson, 2013).

Ques: What is an electron-volt and why is it used?

It is the energy one electron charge gains when it moves through a potential difference of one volt. Since $\Delta U=q\Delta V$, one electron-volt equals $1.602\times 10^{-19}$ J. Energies of electrons in solids are of this size, so the unit is natural for semiconductor work: the band gap of silicon is about 1.12 eV, the thermal energy $k_{B}T$ is about 0.0259 eV at room temperature (computed), and visible photons carry about 1.8 to 3.1 eV. Converting between electron-volts and volts simply means dividing by the electron charge, which is why the thermal voltage $k_{B}T/q$ is about 25.9 mV at 300 K (computed). Reference: B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed. (Pearson, 2015); D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. (Wiley, 2014).

References

1. D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, 10th ed. Hoboken, NJ: Wiley, 2014.
2. W. H. Hayt Jr., J. E. Kemmerly and S. M. Durbin, Engineering Circuit Analysis, 8th ed. New York: McGraw-Hill, 2012.
3. D. J. Griffiths, Introduction to Electrodynamics, 4th ed. Boston: Pearson, 2013.
4. W. H. Hayt Jr. and J. A. Buck, Engineering Electromagnetics, 8th ed. New York: McGraw-Hill, 2012.
5. B. G. Streetman and S. K. Banerjee, Solid State Electronic Devices, 7th ed. Boston: Pearson, 2015.
6. S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. Hoboken, NJ: Wiley, 2007.