Electrostatics

8.1 Introduction

• Gauss' law: the flux of the net electric field through a closed surface equals the net charge enclosed divided by $\varepsilon_0$, i.e. $\oint\vec{E}\cdot\vec{ds} = \dfrac{q}{\varepsilon_0}$.
• It is a tool for finding $E$ whenever the charge distribution has enough symmetry — spherical, cylindrical or planar.
• Standard steps the board expects: identify the charge density (linear, surface, volume); choose a Gaussian surface matching the symmetry; write the flux from Gauss' law; write the flux again by direct calculation using the geometry; equate the two and solve for $E$.

8.2 Application of Gauss' law

• A uniformly charged sphere is equivalent to a point charge at its centre for all external points.
• For the wire, the flat end caps contribute nothing ($\theta = 90^\circ$, $\cos\theta = 0$); all the flux passes through the curved surface, area $2\pi rl$.
• For the sheet, the curved surface contributes nothing; both end caps contribute $EA$ each, so $\phi = 2EA$.
• Distance dependence — the classic marks-loser: sphere $E\propto \dfrac{1}{r^2}$, wire $E\propto\dfrac{1}{r}$, sheet $E$ independent of $r$.
• In a medium of dielectric constant $k$ every one of these is divided by $k$: e.g. $E = \dfrac{\lambda}{2\pi k\varepsilon_0 r}$ outside a charged cylinder.
• Just outside any charged conductor, $E = \dfrac{\sigma}{\varepsilon_0 k}$, normal to the surface, and the outward mechanical force per unit area is $f = \dfrac{1}{2}\varepsilon_0 kE^2$.

8.3 Electric potential and potential energy

• Electrostatic potential energy is the work done against electrostatic forces to bring a system of charges to a given configuration from infinite separation.
• Because the Coulomb force is conservative, $\Delta U$ and hence the work done depends only on the end points, never on the path.
• For a fixed charge $Q$ and a test charge $q_0$ moved from $r_1$ to $r_2$: $W_{AB} = \Delta U = \dfrac{Qq_0}{4\pi\varepsilon_0}\left(\dfrac{1}{r_2}-\dfrac{1}{r_1}\right)$. Taking $U=0$ at $r=\infty$ gives $U = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r}$.
• Potential $V = \dfrac{U}{q}$; potential difference $V_2 - V_1 = \dfrac{W}{q_0}$, the work per unit charge. Potential is a scalar.
• Potential at a point A is the work done in bringing unit positive charge from infinity to A.
• Relation with field: $dV = -E\,dx$, so $E = -\dfrac{dV}{dx}$ — the field is the negative gradient of the potential.
• Zero of potential is arbitrary but must be fixed: infinity for point charges and localised distributions, earth for circuits.
• Units: joule for energy; $1\ \text{eV} = 1.6\times10^{-19}\ \text{J}$ is the kinetic energy an electron gains across a potential difference of 1 volt.
• Trap: $E=0$ at a point does not force $V=0$ there (midpoint of two equal like charges), and $V=0$ does not force $E=0$ (equatorial plane of a dipole).

8.4 Potential due to a point charge, a dipole and a system of charges

• Point charge: $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$, spherically symmetric, independent of the direction of $\vec{r}$, and zero at infinity. A positive charge gives positive potential, a negative charge negative potential.
• Short dipole, general point at distance $r$ and angle $\theta$ to the axis: $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p\cos\theta}{r^2}$, obtained by binomial expansion keeping terms of first order in $\dfrac{l}{r}$.
• Axial point ($\theta = 0^\circ$): $V = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^2}$, the maximum. Equatorial point ($\theta = 90^\circ$): $V = 0$, the minimum, so the equatorial plane is a zero-potential equipotential plane and no work is done moving a charge anywhere in it.
• System of charges: $V = \dfrac{1}{4\pi\varepsilon_0}\left(\dfrac{q_1}{r_1}+\dfrac{q_2}{r_2}+\cdots+\dfrac{q_n}{r_n}\right)$ — an algebraic sum with signs, replaced by an integral for a continuous distribution.
• Standard comparison: for a single point charge $F\propto\dfrac{1}{r^2}$, $E\propto\dfrac{1}{r^2}$ but $V\propto\dfrac{1}{r}$; for a short dipole $V\propto\dfrac{1}{r^2}$.

8.5 Equipotential surfaces

• An equipotential surface is one on which the potential is the same at every point.
• No work is done in moving a test charge anywhere on an equipotential surface, since $V_P = V_Q$ makes $W_{QP} = 0$.
• $\vec{E}$ is always normal to the equipotential surface. If it were not, it would have a component along the surface, and moving a charge against that component would need work — contradicting the definition.
• Equipotential surfaces never intersect: at an intersection there would be two directions of $\vec{E}$ at one point, which is impossible.
• Standard shapes to describe: concentric spheres about an isolated point charge; coaxial cylinders about a line charge; equally spaced parallel planes perpendicular to the field for a uniform field; closed curves that are symmetric about the perpendicular bisector for a dipole (with the equatorial plane as the zero surface); the surfaces between two plane metallic sheets, or between a sheet and a charged sphere, following the electrodes.
• Like lines of force, they picture both the direction of the field (normal to them) and its magnitude (closer surfaces mean a stronger field).

8.6 Electrostatic potential energy of two point charges and of a dipole

• Definition: the total work done in assembling the system of charges by bringing them from infinity to their present positions.
• Two charges: $U = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r_{12}}$ — bringing the first charge in costs nothing, bringing the second costs (potential due to the first) $\times q_2$.
• $N$ charges: $U = \dfrac{1}{4\pi\varepsilon_0}\displaystyle\sum_{\text{all pairs}}\dfrac{q_jq_k}{r_{jk}}$; for three charges there are three pairs, for four charges six pairs.
• Single charge in an external field: $U = qV(\vec{r})$. Two charges in an external field: $U = q_1V(\vec{r_1}) + q_2V(\vec{r_2}) + \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r_{12}}$ — students routinely forget the last, mutual, term.
• Dipole in a uniform field: it feels a torque $\vec{\tau} = \vec{p}\times\vec{E}$ of magnitude $pE\sin\theta$ but zero net force, so it rotates without translating. Torque is maximum ($pE$) when the dipole is perpendicular to the field, zero when it is parallel or antiparallel.
• Work done by the external torque in turning the dipole: $W = pE(\cos\theta_1 - \cos\theta_2)$; this is stored as potential energy.
• Zero-level convention: choosing $U = 0$ at $\theta_0 = 90^\circ$ gives $U = -pE\cos\theta$; choosing $U=0$ at $\theta_0 = 0^\circ$ gives $U = pE(1-\cos\theta)$. State the convention you use.
• Board exercise result worth remembering: for $-q$, $+Q$, $-q$ placed at equal spacings on a straight line, zero total potential energy requires $Q:q = 1:4$.
• Flux check: for a Gaussian sphere, only the charge actually inside contributes, so a sphere of radius $4a$ centred at the origin enclosing only $-4Q$ has flux $-\dfrac{4Q}{\varepsilon_0}$ regardless of the charge sitting outside at $5a$.

8.7 Conductors and insulators, free and bound charges

• Free charges: electrons in the outermost shells of a metal, which drift opposite to an applied field. Bound charges: nuclei and inner-shell electrons, immobile in their positions. In electrolytic conductors positive and negative ions carry the charge, with restricted movement.
• Electrostatic shielding: since $E=0$ inside a hollow conductor, a delicate instrument placed inside one is protected from external charges; thin metal foils make such shields. This is why a car is the safest place in a thunderstorm.
• Faraday cage: an enclosure that blocks external electric fields; MRI scan rooms are shielded so that external radio-frequency signals do not mix with the machine's signals.

8.8 Dielectrics and electric polarisation

• Dielectrics are insulators that can store electrical energy: placed in an external field, their positive and negative charges are displaced in opposite directions and the molecules develop a net dipole moment. This is polarisation. Examples: glass, wax, water, wood, mica, rubber, stone, plastic.

• Polarisation of a non-polar dielectric: the external field $E_0$ pulls the positive centre along the field and the negative centre against it, the molecule gets distorted, and displacement stops when the field force is balanced by the restoring force inside the molecule. Each molecule becomes a tiny dipole; the induced moments add to a net moment.
• Polarisation of a polar dielectric: the permanent moments are randomly oriented by thermal agitation, giving zero net moment; the applied field aligns them, so the slab acquires a net moment along the field. The degree of alignment is set by the competition between the applied field and the field of the induced dipoles.
• Both kinds develop a net dipole moment in a field. Alignment moves charges only over distances less than an atomic diameter, so no current flows.
• Polarisation $\vec{P}$ is the dipole moment per unit volume; for linear isotropic dielectrics $\vec{P} = \chi_e \vec{E}$, where $\chi_e$ is the electric susceptibility ($\chi_e = 0$ for vacuum).
• Reduction of field: the aligned dipoles leave an apparent sheet of positive charge on one face and negative on the other. These induced surface charges set up a polarisation field $E_p$ opposing $\vec{E}$, so the net field inside is $E' = E - E_p$, smaller than the applied field.
• The induced moment disappears when the field is removed; it is what makes a charged comb attract uncharged bits of paper.
• Dielectric strength is the field at which a dielectric breaks down, gets ionised and begins to conduct. The dielectric constant of a conductor is infinite; that of air is 1.

8.9 Capacitors and capacitance, combinations

• A capacitor is a pair of conductors carrying equal and opposite charges, separated by a dielectric, used to store electrical energy (unlike a resistor, which only dissipates).
• Capacitance $C = \dfrac{Q}{V}$ is the ability of a conductor to store charge. It depends on size, shape and separation of the conductors, and on the medium — not on $Q$ or $V$.
• SI unit farad: a capacitor has capacitance 1 farad if the potential difference across it rises by 1 volt when 1 coulomb of charge is given to it. Dimensions $[\text{M}^{-1}\text{L}^{-2}\text{T}^{4}\text{A}^{2}]$. Practical units are $\mu\text{F}$, nF and pF.
• Principle of a capacitor: bringing an earthed metal plate $P_2$ near a charged plate $P_1$ induces a negative charge on the near face of $P_2$ (which lowers the potential of $P_1$) and a positive charge on the far face (which flows to earth). The net potential of $P_1$ falls, so its capacitance $C = \dfrac{Q}{V-V_1}$ rises.

• In series, the capacitor of smallest capacitance carries the largest potential difference.
• For two capacitors in series the working form is $C_S = \dfrac{C_1C_2}{C_1+C_2}$.
• Mixed networks: reduce the innermost parallel groups first, then the series chain. For $C_1$ and $C_2$ in parallel, in series with $C_3$: $C = \dfrac{(C_1+C_2)C_3}{C_1+C_2+C_3}$.

8.10 Parallel plate capacitor, with and without a dielectric

• The two plates each of area $A$ are held a distance $d$ apart; one is charged and the other earthed. In the outer regions the two plate fields cancel ($E=0$); between them they add, giving $E = \dfrac{\sigma}{\varepsilon_0} = \dfrac{Q}{A\varepsilon_0}$, directed from the positive to the negative plate.
• With $V = Ed$ this gives $C_0 = \dfrac{\varepsilon_0 A}{d}$ — capacitance rises with plate area and falls with separation.
• Filling the gap completely with a dielectric of constant $k$: $C = kC_0$, so $k = \dfrac{C}{C_0}$.
• A slab of thickness $t<d$: $C = \dfrac{\varepsilon_0 A}{d-t+\dfrac{t}{k}}$. A conducting slab ($k=\infty$) gives $C = \dfrac{d}{d-t}C_0$, always an increase, and independent of where the slab is placed.
• Standard result asked as an objective: a slab of thickness $\dfrac{3d}{4}$ gives $C = \dfrac{\varepsilon_0 A}{d}\left(\dfrac{4k}{k+3}\right)$.
• Stacked slabs behave as capacitors in series; slabs side by side over parts of the area behave as capacitors in parallel.

• Read the reverse of the right-hand column for the standard question "the battery is disconnected and then the dielectric is removed": $C$ decreases, $Q$ stays constant, and $V$, $E$ and $U$ all increase $k$ times.
• For a charged and isolated capacitor whose plate separation is then increased: $Q$ stays constant, $C$ decreases and $V$ increases.
• Displacement current (8.11): in the dielectric between the plates there are no free electrons, but the bound charges move, giving $i_d = Ak\varepsilon_0\dfrac{dE}{dt}$, equal to the conduction current $i_c = \dfrac{dq}{dt}$ in the leads. A displacement current exists at any point in space where $E$ varies with time.

8.12 Energy stored in a capacitor

• Charging a capacitor means transferring electrons from one plate to the other against the Coulomb forces of both plates, and the opposing force grows as the charge grows.
• Work to transfer a further $dq$ is $dW = \dfrac{q'}{C}dq$; integrating from 0 to $Q$ gives $W = \dfrac{1}{2}\dfrac{Q^2}{C}$, stored as electrostatic energy in the field between the plates.
• Three equivalent forms: $U = \dfrac{1}{2}\dfrac{Q^2}{C} = \dfrac{1}{2}CV^2 = \dfrac{1}{2}QV$.
• Energy density $u = \dfrac{1}{2}\varepsilon_0 kE^2$ in $\text{J m}^{-3}$; multiply by the volume of dielectric to get the energy contained in it.
• The battery does work $QV$ but only $\dfrac{1}{2}QV$ is stored; the remaining half is lost as heat in the connecting wires and the battery. Hence energy stored : energy dissipated during charging is $1:1$.
• Sharing charge between two capacitors always loses energy: $\Delta U = \dfrac{1}{2}\dfrac{C_1C_2}{C_1+C_2}(V_1-V_2)^2$, lost as heat and electromagnetic radiation.

8.13 Van de Graaff generator

• Purpose: to develop very high potentials, of the order of $10^{7}$ volt, whose large fields accelerate charged particles (electrons, protons, ions) to energies of the order of 10 MeV. Designed by Van de Graaff in 1931.
• Three principles: corona discharge (the action of sharp points); charge given to a hollow conductor moves entirely to its outer surface and spreads uniformly; continuously supplying charge to an insulated conductor raises its potential indefinitely.
• Labels for the diagram: $P_1$, $P_2$ pulleys; BB insulating conveyor belt; A spray brush; C collector brush; D dome-shaped hollow metallic conductor; E evacuated accelerating tube; I electrode (ion source end); P high-voltage DC supply; S steel vessel filled with nitrogen at high pressure; M earthed metal plate.
• Working in one line: the motor drives the belt, brush A sprays positive charge on it, the belt carries the charge up to brush C which delivers it to the dome D, the charge spreads on the outer surface of D and its potential rises enormously, and positive ions introduced at the top of tube E are repelled by electrode I and accelerated down the tube to strike a target.
• Nitrogen at high pressure (with a little freon) inside the steel vessel prevents leakage of charge from the dome.
• Uses: disintegration of nuclei, production of radioactive isotopes, study of nuclear structure and of nuclear reactions, and accelerating electrons to sterilise food and process materials.