Conductors in electrostatics

In a conductor electrons are free to move.  If they are acted on by a force, they will accelerate in the direction of the force.  If a conductor is placed into an external electric field, a force F = -eE acts on each free electron.  Electrons accelerate and gain velocity in a direction opposite to the field.  Soon electrons will pile up on the surface on one side of the conductor, while the surface on the other side will be depleted of electrons and have a net positive charge.  These separated negative and positive charges on opposing sides of the conductor produce their own electric field, which opposes the external field inside the conductor and modifies the field outside.

When enough electrons have piled up on one side and enough positive charge has been left on the other side, then the field produced by these charges exactly cancels the external field inside the conductor, and electrons inside the conductor no longer experience a force.  This is the case in the picture shown above.  The inside of the conducting sphere is field-free, while the previously constant external field outside has been modified.

In static equilibrium the inside of a conductor is field free.  If it were not, electrons would move and distribute themselves, so as to cancel out the field.  The inside of a conductor can not contain any net charge.  Such charges would produce a field inside the conductor, and electrons would move and cancel out the field and neutralize the charge.  Any excess charge on a conductor must therefore reside on the surface.  The field just outside the conductor at the surface must be perpendicular to the surface.  If it were not, electrons would redistribute themselves to cancel out the field.  The strength of the electric field on the surface of a conductor can be found by applying Gauss' law.

The electric flux through the surface shown in the figure above is F = Q_inside/e₀ = sA/e₀, where s is the surface charge density and A is the area of the conductor's surface inside the Gaussian surface shown.  The flux through the sides of the Gaussian surface is zero, since E is perpendicular to the surface of the conductor.  The flux through the bottom of the Gaussian surface is zero since the electric field inside a conductor is zero.  The total flux through the Gaussian surface therefore equals the flux through the top, and we have F = EA.  We therefore have for the strength of the electric field near the surface of a conductor

E = s/e₀.

The surface of any conductor is an equipotential surface.  The field is everywhere perpendicular to the surface.  No work is being done moving a charge along on the surface.  The surface of a spherical conductor with radius R, carrying a charge Q is at a potential V = k_eQ/R.  If we have two spherical conductors with radii R₁ and R₂, respectively, at the same potential V, they carry charges Q₁ = R₁V/k and Q₂ = R₂V/k, respectively.  The electric fields near their surfaces are E₁ = k_eQ₁/R₁² = V/R₁ and E₂ = k_eQ₂/R₂² = V/R₂ respectively.  The smaller the radius, the larger is the electric field.  In general, near the surface of a conductor, the field is largest in places with the smallest local radius of curvature.

Very strong fields are found near sharp conducting tips.  Air molecules will be stripped of electrons if the field becomes too large (~3´10⁶ V/m).  The free electrons accelerate and collide with other molecules to make more ions and electrons.  A plasma forms between the conductor and the ground and the conductor discharges.  This is called a corona discharge.

A device that makes use of the strong field near a tip is the field ion microscope.  The field ion microscope has a sharp tip with a local radius of curvature of ~10 - 100 nm.  This tip faces a phosphor screen.  Under vacuum, a potential difference is established between the tip and the screen.  The tip is held at the more positive potential.  A small amount of inert gas is admitted, and gas atoms near the tip are ionized. Electrons are ripped off these atoms.  The positively charged ions are accelerated by the intense electric field along a straight line toward the phosphor screen, where they are detected by converting their kinetic energy into light.  Each point on the tip maps into a different point on the screen, so that a magnified, image of the tip can be viewed.  Since the tip is only 10 - 100nm in radius, one can achieve atomic resolution.  A typical field ion microscope image of a 'single crystal' tungsten tip is shown below:

The bright spots correspond to positions on the tip where the electric field is particularly high, i.e. where the local radius of curvature is particularly small.  This happens near atoms, so the microscope images the position of atoms in the tip.

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Summary:

In electrostatic equilibrium a conductor has the following properties.

The entire conductor is at the same potential.  There is no field inside the conductor.  A cavity inside a conductor, completely surrounded by conducting material, also is free of electric fields, if it does not contain any net charge itself.  A conductor shields its interior from any outside electric fields.  Even if there are holes in the surface, the electric field does not penetrate very far.  A rule of thumb is that the electric field falls to zero over a distance approximately equal to the diameter of the hole.

In the diagram above, the field only penetrates a small distance through the holes into the box with conducting walls.

Why are you safest inside your car during a thunderstorm?

Problem:

Capacitors

A capacitor is a device for storing charge.  No single electronic component plays a more important role today than the capacitor.  This device is used to store information in computer memories, to regulate voltages in power supplies, to establish electrical fields, to store electrical energy, to detect and produce electromagnetic waves, and to measure time.

Any two conductors separated by an insulating medium form a capacitor.  A parallel plate capacitor consists of two plates separated by a thin insulating material known as a dielectric.  One plate of the capacitor holds a positive charge Q, while the other holds a negative charge -Q.  The charge Q on the plates is proportional to the potential difference V across the two plates.  The capacitance C is the proportional constant,

Q = CV, C = Q/V.

C depends on the capacitor's geometry and on the type of dielectric material used.  The capacitance of a parallel plate capacitor with two plates of area A separated by a distance d and no dielectric material between the plates is

C = e₀A/d.

(The electric field is E = s/e₀. The voltage is V = Ed = sd/e₀. The charge is Q = sA. Therefore Q/V = sAe₀/sd = Ae₀/d.)  The SI unit of capacitance is Coulomb/Volt = Farad (F).  Typical capacitors have capacitances in the picoFarad to microFarad range.

The capacitance tells us how much charge the device stores for a given voltage.  A dielectric between the conductors increases the capacitance of a capacitor.  The molecules of the dielectric material are polarized in the field between the two conductors.  The entire negative and positive charge of the dielectric is displaced by a small amount with respect to each other.  This results in an effective positive surface charge on one side of the dielectric and a negative surface charge on the other side of the dielectric.  These effective surface charges on the dielectric produce an electric field, which opposes the field produced by the surface charges on the conductors, and thus reduces the voltage between the conductors.  To keep the voltage up, more charge must be put onto the conductors. The capacitor thus stores more charge for a given voltage.  The dielectric constant k is the ratio of the voltage V₀ between the conductors without the dielectric to the voltage V with the dielectric, k = V₀/V, for a given amount of charge Q on the conductors.

In the diagram above, the same amount of charge Q on the conductors results in a smaller field between the plates of the capacitor with the dielectric.  The higher the dielectric constant k, the more charge a capacitor can store for a given voltage.  For a parallel-plate capacitor with a dielectric between the plates, the capacitance is

C = Q/V = kQ/V₀ = ke₀A/d.

The static dielectric constant of any material is always greater than 1.

Typical dielectric constants:

Energy stored in a capacitor

The energy stored in a capacitor is equal to the work done in separating the charges on the conductors.  The more charge is already stored on the plates, the more work must be done to separate additional charges, because of the strong repulsion between like charges.  At a given voltage, it takes an infinitesimal amount of work dW = VdQ to separate an additional infinitesimal amount of charge dQ.  (The voltage V is the amount of work per unit charge.)  We can write dW = (Q/C)dQ, since V = Q/C.  To find the total work done in charging the capacitor we integrate,

.

The energy stored in a capacitor therefore is

U = (1/2)(Q²/C).

Using Q = CV we can also write

U = (1/2)CV².

Problem:

For any insulator, there is a maximum electric field that can be maintained without ionizing the molecules.  For a capacitor this means that there is a maximum allowable voltage that that can be placed across the conductors.  This maximum voltage depends the dielectric in the capacitor.  The corresponding maximum field is called the dielectric strength of the material.  For stronger fields, the capacitor 'breaks down' (similar to a corona discharge) and is normally destroyed.  Most capacitors used in electrical circuits carry both a capacitance and a voltage rating.  This breakdown voltage V_b is related to the dielectric strength E_b.  For a parallel plate capacitor we have V_b = E_bd.

Capacitors in parallel

Consider two capacitors in parallel as shown below.

When the battery is connected, electrons will flow until the potential of point A is the same as the potential of the positive terminal of the battery and the potential of point B is equal to that of the negative terminal of the battery.  Thus, the potential difference between the plates of both capacitors is V_A - V_B = V_bat.  We have C₁ = Q₁/V_bat and C₂ = Q₂/V_bat, where Q₁ is the charge on capacitor C₁, and Q₂ is the charge on capacitor C₂.  Let C be the equivalent capacitance of the two capacitors in parallel, i.e. C = Q/V_bat, where Q = Q₁ + Q₂.  Then C = (Q₁ + Q₂)/V_bat = C₁ + C₂.  For capacitors in parallel, the capacitances add.  For more than two capacitors we have

C = C₁ + C₂ + C₃ + C₄+….

Capacitors in series

Let Q represent the total charge on the top plate of C₁, which then induces a charge -Q on its bottom plate.  The charge on the bottom plate of C₂ will be -Q, which in turn induces a charge +Q on its top plate as shown.

Let V₁ and V₂ represent the potential differences between plates of capacitors C₁ and C₂, respectively.  Then V₁ + V₂ = V_bat, or (Q/C₁) + (Q/C₂) = Q/C, or (1/C₁) + (1/C₂) = 1/C.  For more than two capacitors in series we have

where C is equivalent capacitance of the two capacitors.

Electrostatic air cleaners

Electrostatic air cleaners remove dust, soot, and ash particles from normal air.  Each dust, soot, or ash particle has mass.  The air exerts two types of forces on the particles, the buoyant force and the viscous drag force.  Since the particles are denser than air, the buoyant force alone cannot support the particles.  The viscous drag force keeps the particles from descending quickly.  For small particles the terminal velocity can be lower than 1mm/s.  The drag force opposes relative motion between the particles and the air, and moving air tends to carry the particles along with it.  The slightest upward breeze can keep the dust, soot, and ash particle aloft.

Electrostatic air cleaners use electrostatic forces to pull these particles from the air.  A typical air cleaner gives each particle a negative charge and then collects it on a positively charged surface.

How does a dust grain become negatively charged?

The air cleaner uses a corona discharge to give the dust grain a negative charge.  A power supply does work maintaining a potential difference of approximately 10000V between the corona wires and the collecting surfaces.  The negatively charged dust flows with the air through the air cleaner.  When it passes a positively charged surface, it experiences an electrostatic force strong enough to overwhelm the viscous drag.  The dust particles quickly leave the air stream and collect on the charged surface.  The air continues on without the dust.  The air cleaner precipitates clumps of dust on its collecting plates, and therefore is called an electrostatic precipitator.  It can accumulate large amounts of dust on its plates without blocking the airflow and it is easy to clean.  When several centimeters of dust have accumulated on the collecting surfaces, it is removed by rapping the plates with a stick.  The sudden blow causes the plates to accelerate rapidly and they leave the dust behind.  It falls in clumps to the bottom of the precipitator, where it’s collected for recycling or disposal.

Ion Generators

Household ion generators are also effective at removing dust and smoke from room air.  These machines resemble electrostatic precipitators, but they have no internal collecting plates.  They use a corona discharges to charge passing molecules and dust grains and then let those charged particles drift into the room.  When the charged particles come near a surface, they polarize the surface and are attracted to it.  Although this method is cheap and effective, it slowly dirties the walls and the furniture.

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Photocopiers

At the heart of the photocopier is a thin layer of photoconductor.  A photoconductor is a solid material through which electrons can move only when it is exposed to light.  In the dark, it is an insulator, in the light, it’s a conductor.  This property allows light to determine the pattern of static electricity on a copying drum and hence the placement of toner on a piece of paper.

The light sensitive component in a photocopier is a metal drum that is covered with a thin layer of photoconductor.  This metal drum is grounded.  The copier coats the photoconductor with electrons, which remain in place as long as the photoconductor is in the dark.  But wherever light strikes the photoconductor, it becomes conductive and allows the electrons to escape through the metal and flow into the ground.  Only the portions of the photoconductor which are not illuminated retain their static electric charge and eventually attract black toner particles.  In that manner, the darkened parts of the photoconductor produce the dark parts of the final copy.

The copier starts by applying a uniform negative charge to the surface of the photoconductor.  This charge is applied by a corotron, a fine wire centered in a half-cylinder of metal.  A power supply pumps electrons onto the fine wire until they are emitted into the air as a corona discharge.  When these electrons approach the photoconductor they polarize it and stick to it.  The photoconductor becomes uniformly charged, with about 10^-7C of negative charge per cm² of surface.  After the charging, the copier exposes the photoconductor to light from the original document.  It uses a lens to cast an image of the original onto the photoconductor’s surface.  Light only hits the photoconductor in certain places, which correspond to the white parts of the original document.  When the exposure is over, the photoconductor carries a charge image of the original document.

To develop this charge image into a visible one, the photocopier exposes the photoconductor to charged toner particles.  The toner is a fine insulating plastic powder.  A spinning brush with extraordinarily soft bristles wipes toner particles out of their storage tray onto the photoconductor.  During this transfer, the toner particles become positively charged so that they stick to the negatively charged portions of the photoconductor.  The photoconductor now carries a black image of the original document.  But to create a copy, this black image must be transferred to paper.  To begin this transfer, the copier illuminates the photoconductor with a charge erase lamp so that the photoconductor’s negative charge escapes into the metal.  The toner remains in place but it is only very weakly attached.  The copier than transfers the toner to a nearby sheet of paper by applying negative charge to the paper’s back.  The positively charged toner is attracted to the negatively charged paper and the two leave the photoconductor together.  The copier then heats and presses the copy, permanently fusing the toner into the paper.  Sometimes, when a copier jams, you may remove a sheet before it has been fused.  The image looks completely normal but wipes off when you touch it because it’s held in place only by electrostatic forces.  Once the image has been transferred to the paper the drum is cleaned.  The photoconductor is then ready to be used again.

A laser printer is also a photocopier device, but it uses a laser beam to write a charge image directly onto its photoconductor drum.   Wherever laser light hits the drum, charge flows through the photoconductor.  A computer in the printer turns the laser on and off as it systematically constructs the charge image, one dot at a time.  The photoconductor and the toner supply are contained in a single disposable cartridge.

Other links concerning electrostatics:

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