## Wednesday, 1 June 2011

### Physics : Electrostatics

Electrostatics
We say two charge (+/-) exists, which like charge repels and unlike charge attracts each other.
Basic atomic model: matters are made up of atoms, in which protons (p+, +ve charged) and neutrons are in the tiny nucleus and electron (e-, -ve charged) orbits the nucleus.
An atom has equal number of p+ and e- so it’s electrically neutral. It becomes charged when its electron transferred to other place (or receive extra electron), (while proton do not move), then it will be overall charged.
Conductor has free electron to flow inside while insulator does not.
Charging an insulator:
1)       When you rub the insulator, electron transfer from one material to another material, so it becomes charged.
2)       When it’s put near a charged object, the electron cloud in each atom is distorted so that the charge opposite to the charged object is induced in the near side while the far side has the same type of charge with the charged object. As a result the electrostatic force between the near side and the object is greater than the electrostatic repulsion between the far side and the object, so it’s attracted to the charged object).
Charging a conductor:
1)       When it’s put near a conductor, the free electrons are attracted/repelled to one side of the conductor. As a result the object become one side positively charged and another side negatively charged. We say the charge is induced. Note that when the charged object is removed, the charge tends to neutralize and disappear.
2)       When it’s touched by another charged conductor, the charge is shared by the two conductors (the same type of charge), so in the next moment they repel under electrostatic repulsion if they’re not fixed. We say the charge is produced by sharing.
Earthing:
Earth is considered as a large conductor so that all charge will be transferred to the Earth under earthing. For example when the conductor has positive charge, under earthing the charge is “neutralized” by the charge of Earth, but in fact electron from the Earth flows into the conductor. So there’s a common way to produce a net charge on an conductor: induce a charge on an conductor -> earth one side of the conductor (it affects one side only since the change in another side is attracted by the charged object) -> remove the charged object.
EHT supply: This apparatus provide up to 1kV ~ 5kV of voltage.
Van de Graaff generator: when it’s switched on, the friction in its rubber band produces charges and the negative charges are trapped in the metal dome at the top. It provides up to 105V voltage.

Gold-leaf electroscope: It’s an apparatus to find the existence of charged objects: when a charged object is put near it, charged is induced at the top of the electroscope while opposite charge is induced at the bottom (brass strip), when it’ll repel due to electrostatic repulsion.
Electrostatic hazard may occur like the charge stored in type may be electric shock to passengers or cause sparks for explosion. A metal chain is left on the ground to earth the charge or a conductor tires are used.
Also, it may cause inconvenience to daily life like difficulty in separating paper, or attracting dusts when discs are rubbed by the needle.
Applications
1)       Precipitation: under strong E-field dusts are charged and attracted to the wall.
2)       Photocopying: The drum is positively charged, then the image of the photo is shone on the drum (white surface -> transparence -> light shone on charged paper -> charge disappeared in that area), then negatively charged toner sticks on the drum and stick on the paper. The paper is heated and rolled to give a permanent image.
3)       Electrostatic separator by a E-field. Conductors are deflected while insulators are sticked on the positively charged rotor..
The unit of charge is coulomb (C) which is the amount of charge passing through the point in 1A in 1s.
Coulomb’s law
It states that the electrostatic force between two point charges are acting along the line joining the two charge, directly proportional to their charge while inversely proportional to the square of distance between them. Mathematically F = Qq/r2 (1/4πε0), where ε0 is the permittivity of free space, which is equal to 8.85*10-12 C2N-1m-2. Positive force implies repulsive while negative force implies attractive forces.
Electric field is produced by charged objects, where densities of field lines represent the field intensity. Charged object experienced a electric force under electric field.

The electric field flows from positive to negative charge, which is the direction of force experienced by a positive charge (the direction of force in a curved E-field is given by the tangent of the E-field lines at that point.). There’s no closed-loop field line (otherwise W.D. to move around the loop is negative which contradicts the conservation of energy.) The electric field strength E is defined as the magnitude of force when a unit charge (positive) is placed at that point. The unit is NC-1.
We can say that F = Eq = Qq/r2 (1/4πε0) = Eq, so E = Q/r2 (1/4πε0) which is independent to the point charge experiencing the electric force. The E-field for a point charge is radially inward/outwards.
We can also use a pair of parallel plate to produce a uniform E-field (inside the parallel plates). The electric field strength is determined by surface charge density on the plates:
E = σ / ε0 = Q/Aε0
In fact, we can use the electric field to deflect a hanged point charged mass to find its charge/mass/electric field strength by measuring the deflected angle.
For example, a charged point of charge q and mass m is hung by a light string, and a deflection θ is measured under a electric field E. We can get Eq = F = mgtan θ.
Electrical potential energy
Work has to be done when a charge is brought to another charge distribution due to the electrostatic force between them. Consider the integrated version of W = Fs, we have the electrical potential energy U = ∫ FE dr = ∫ Qq/r2 (1/4πε0) dr = Qq/4πε0 ∫ dr/r2 = -Qq/4πrε0 + C. However, we have taken the reference point at infinitely far as zero. Then U is defined as the work done to bring a charge from infinitely far to the required point. Therefore C = 0 while the sign is reversed, which is U = Qq/4πrε0. By definite integral we have U = Qq/4πrε0. Bringing a point charge from A to B is given by ΔU = UB-UA = Qq/4πε0(1/rB-1/rA).
Significance:
1)       When two unlike charges are brought together, its ΔU is negative, we have energy is released and is spontaneous; when two like charges are brought together, its ΔU is positive and work done has to be provided.
2)       U of a charge infinitely far is zero.
3)       ΔU is independent of its path, like work done in mechanics.
In parallel plate, F = qE is also valid, so a shift d of point charge in electric field perpendicular to the plates have a work done qEd. Also in collision of two charged mass, we balance the E.P.E. and K.E. to find the nearest distance between them.
Electrical potential is another value with no absolute value, and we define electrical potential V at B relative to A is the work done to bring a unit positive charge from A to B. When we take infinitely far as the reference point, we have V = U/q = Q/4πrε0. When Q is positive, V is positive while V is negative when Q is negative. We can conclude that positive charges tend to move to region of lower potential while negative charge tends to move to region of higher potential (both tends to zero and proportional to 1/r).
We can also get ΔU = qΔV, we call ΔV as the potential difference (p.d.), for example when a positive charge moves to higher potential, then it gains E.P.E.
Consider the potential in a parallel plate: we assume the negative plate is earthed, i.e., V of the negative plate is zero. V = U/q = qEd/q, so we have E = V/d. Also consider the potential at a point of distance x from the negative plate, V = Ex, or V0(d-x/d), where V0 is the potential of the positive plate.
Equipotential lines (surface) are a line (surface) in which at points in the line (surface) have the same potential. In parallel plates the equilpotential lines (surfaces) are parallel to plates while it’s a equidistant circle (sphere) in a point charge. It must be perpendicular to E-field, otherwise the non-zero component of E-field along the equilpotential line cause non-zero work done to move along the equilpotential lines which is contradictory.
When there’s no parallel plates, E = v/d is still valid but we say E = -dV/dr which is proved by the basic definition of E and V. It also checked the validity of E = -dV/dr. When the slope of potential change is constant, the electric field inside a parallel plate is also constant.

Lastly, the negative sign of E = -dV/dr shows that the electric field flows in the direction of decreasing potential.
Gravitation VS electrostatics
We know that both gravitational force and electrostatic force are the basic form of force, they have many similarities:
1)       Force: FG = GMm/r2, FE = Qq/r2 (1/4πε0)
2)       Field: g = FG/m = GM/r2, E = FE/q = Qq/r2 (1/4πε0)
3)       Potential energy: UG = ∫ FGdr = -GMm/r, UE = ∫ FEdr = Qq/r (1/4πε0)
4)       Potential: VG = UG/m = -GM/r, VE = UE/q = Q/r (1/4πε0)
So we know that U and V are scalar and it can be added despite the direction of charge producing this potential or E.P.E., while F and E are vector and vector sum is considered.