TOPIC 1 ELECTROSTTICS PHY11 Electricity an Magnetism I Course Summary Coulomb s Law The magnitue of the force between two point charges is irectly proportional to the prouct of the charges an inversely proportional to the square of the istance between them. q F = r ˆ Principle of Linear Superposition F = F + F + + F TOPIC ELECTRIC FIELDS Electric fiel is force per unit (test) charge. Fiel ue to a point charge Force exerte on a charge q by a fiel 1 1 13 1N E = F = qe Conuctors Uner static conitions, there is no net macroscopic fiel within the material of a conuctor. Uner static conitions, the electric fiel at all points on the surface of a conuctor is normal to that surface. Uner static conitions, all the (unbalance) electric charge resies on the surface of the conuctor. The fiel ue to a continuous charge istribution is q E = r ˆ 4πε r Electric Dipoles The electric ipole moment of a pair of equal an opposite charges separate by a istance is p = In an electric fiel, the torque on an electric ipole is τ = p Ε an the potential energy of the ipole is U = pe cos θ = p E TOPIC 3 GUSS S LW Electric Flux The electric flux (or number of fiel lines) intercepte by an area is given by Φ = E = E cos θ r ˆ PHY11 E&M I Summary 1 CNB
Gauss s Law The electric flux through a close surface is equal to the total charge containe within that surface ivie by ε. Ei = ε TOPIC 4 Potential Electric Potential The change in electrostatic potential is the change in potential energy per unit charge U q The potential at a point is the external work require to bring a positive unit charge from a position of zero potential to the given point, with no change in kinetic energy. In general VB V = E s The electrostatic potential ue to a point charge is The electrostatic potential ue to a charge conucting sphere of raius R is for r > R 4 πε r for r < R 4 πε R By linear superposition, the total potential at a point ue to a system of charges is the sum of the potentials ue to the iniviual charges. ll points within an on the surface of a conuctor in electrostatic equilibrium are at the same potential. Calculating Electric Fiel from Potential In one imension, the fiel can be erive from the potential istribution using V E = r In vector form, E = V where =,, x y z Potential Energy of a System of Charges The potential energy of a charge q at an electrostatic potential V is U = qv The potential energy of a pair of charges, 1 an, separate by a istance r is 1 U =. 4 πε r Note that this is the energy of the system, not of each charge iniviually. B PHY11 E&M I Summary CNB
continuous charge istribution will have potential energy ue to each element of charge experiencing the fiel ue to the rest of the charge. For example, a conucting sphere of raius R carrying a charge tot will have potential energy tot q 1 tot U = q = (Note the factor ½!) 4πε R 4πε R Energy of Moving Charges For a freely moving charge, the gain in kinetic energy is equal to the loss in potential energy K = q V TOPIC 5 Capacitors Capacitance Capacitors store electric charge, with the capacitance C equal to the charge store per unit potential ifference, so = CV The capacitance of a parallel plate capacitor is ε C = Capacitors in parallel have a combine effective capacitance C = C + C + + C 1 N Capacitors in series have a combine effective capacitance given by 1 1 1 1 = + + + C C1 C C N The energy store in a capacitor (equalling the work one by a battery or other source in charging it) is 1 1 1 U = = CV C The energy ensity in an electric fiel is 1 u = ε E Dielectrics ielectric material reuces the electric fiel by a factor calle the ielectric constant, k. The capacitance is therefore increase by k, e.g. for the parallel plate capacitor kε C = TOPIC 6 Current an Resistance Electric Current Current is the rate of flow of charge I = t (Conventional) current flows from high to low potential. (In a metal, negative electrons move in the opposite irection.) PHY11 E&M I Summary 3 CNB
The current ensity in a wire with charge carriers of charge q an ensity n, with rift velocity v, is I J = = nqv Resistance an Resistivity The resistance of a conuctor epens on the material an its geometry, an is given by V R = I The conuctivity σ an resistivity ρ, which epen only on the material, are efine by 1 J = σ E = E ρ The relationship between resistance an resistivity is ρ R = l The temperature coefficient of resistivity α escribes how resistivity changes with temperature ρ T ρ 1+ α T T ( ) ( ) ( ) Electrical Power The power issipate by a current flowing in a resistive meium is V P = I I R = R Ohm s Law Ohm s law states that, uner certain circumstances, the current flowing through a conuctor is proportional to the potential across it. That is, the electrical resistance is a constant. The Drue Moel of Conuction Charge carriers scatter off the lattice with a mean time between collisions τ. When an electric fiel is applie, the mean rift velocity is hence ee v = τ m m an the resistivity is ρ = ne τ TOPIC 7 Electric Circuits Electromotive Force an Batteries The EMF E of a battery is the work one per unit charge move from negative to positive terminals. It is thus measure in volts. Real batteries have an internal resistance, so their terminal voltage is reuce when current flows, V T = E I r Resistors in Series an Parallel Resistors in series have a combine effective resistance R = R + R + + R 1 N PHY11 E&M I Summary 4 CNB
Resistors in parallel have a combine effective resistance given by 1 1 1 1 = + + + R R1 R R N Kirchhoff s Rules o The junction rule states that: The algebraic sum of the currents entering an leaving a junction is zero. Σ I = o The loop rule states that: The algebraic sum of the changes in potential aroun any close loop is zero. Σ RC Circuits When a capacitor ischarges through a resistor, its charge varies as = e t RC When a capacitor is charge through a resistor, its charge increases as ( ) = e 1 t RC Copies of lecture hanouts an PowerPoint isplays are available from the Web, at http://www.cbooth.staff.shef.ac.uk/phy11e&m/ PHY11 E&M I Summary 5 CNB