Coulomb’s Law Charges with the same sign repel each other, and charges with opposite signs attract each other. The electrostatic force between two particles is proportional to the amount of electric charge that each possesses and is inversely proportional to the distance between the two squared. q1 q2 r by 1 on 2 1,2 1,2 1,2 Coulomb constant: where e0 is called the permittivity constant.

Electric Field Define electric field, which is independent of the test charge, q2 , and depends only on position in space: Electric Field due to a Point Charge Q

Dynamics of a Charged Mass in Electric Field For -Q<0 in uniform E downward: -Q Oscilloscope Ink-Jet Printing vy = at = qE/m t vx >>0 Oil drop experiment http://canu.ucalgary.ca/map/content/force/elcrmagn/simulate/electric_single_particle/applet.html

Electric Field from Coulomb’s Law + - Bunch of Charges Continuous Charge Distribution dq P P k (volume charge) (surface charge) (line charge) Summation over discrete charges Integral over continuous charge distribution http://www.falstad.com/vector3de/

Gauss’s Law: Quantitative Statement The net electric flux through any closed surface equals the net charge enclosed by that surface divided by ε0. How do we use this equation?? The above equation is TRUE always but it doesn’t look easy to use. BUT - It is very useful in finding E when the physical situation exhibits a lot of SYMMETRY.

Charges and fields of a conductor In electrostatic equilibrium, free charges inside a conductor do not move. Thus, E = 0 everywhere in the interior of a conductor. Since E = 0 inside, there are no net charges anywhere in the interior. Net charges can only be on the surface(s). The electric field must be perpendicular to the surface just outside a conductor, since, otherwise, there would be currents flowing along the surface.

Electric Potential Energy of a Charge in Electric Field Coulomb force is conservative => Work done by the Coulomb force is path independent. Can associate potential energy to charge q0 at any point r in space. It’s energy! A scalar measured in J (Joules)

Electric Potential Scalar! taking the same reference point U(r) of a test charge q0 in electric field generated by other source charges is proportional to q0 . So U(r)/q0 is independent of q0, allowing us to introduce electric potential V independent of q0. taking the same reference point [Electric potential] = [energy]/[charge] SI units: J/C = V (volts) Scalar! A positively charged particle produces a positive electric potential; a negatively charged particle produces a negative electric potential. Potential energy difference when 1 C of charge is moved between points of potential difference 1 V

E from V We can obtain the electric field E from the potential V by inverting the integral that computes V from E: Expressed as a vector, E is the negative gradient of V

Electric Potential Energy and Electric Potential positive charge High U (potential energy) Low U High V Low V Electric field direction High V (potential) Low V Electric field direction negative charge High U Low U

Two Ways to Calculate Potential Integrate -E from the reference point at (∞) to the point (P) of observation: Q P r A line integral (which could be tricky to do) If E is known and simple and a simple path can be used, it may be reduced to a simple, ordinary 1D integral. Integrate dV (contribution to V(r) from each infinitesimal source charge dq) over all source charges: q1 q2 q3 q4 P P Q

Capacitance The two conductors hold charge +Q and –Q, respectively. Capacitor plates hold charge Q The two conductors hold charge +Q and –Q, respectively. The capacitance C of a capacitor is a measure of how much charge Q it can store for a given potential difference ΔV between the plates. Expect Capacitance is an intrinsic property of the capacitor. farad (Often we use V to mean ΔV.)

Steps to calculate capacitance C Put charges Q and -Q on the two plates, respectively. Calculate the electric field E between the plates due to the charges Q and -Q, e.g., by using Gauss’s law. Calculate the potential difference V between the plates due to the electric field E by Calculate the capacitance of the capacitor by dividing the charge by the potential difference, i.e., C = Q/V.

Energy of a charged capacitor How much energy is stored in a charged capacitor? Calculate the work required (usually provided by a battery) to charge a capacitor to Q Calculate incremental work dW needed to move charge dq from negative plate to the positive plate at voltage V. Total work is then

Dielectrics between Capacitor Plates + Q - Q free charges Electric field E between plates can be calculated from Q – q. neutral -q +q Polarization Charges ± q The presence of a dielectric weakens the electric field, therefore weaken the potential drop between the plates, and leads to a large capacitance.

Capacitors in Parallel V is common Equivalent Capacitor: where

Capacitors in Series q is common Equivalent Capacitor: where

Electric Current Magnitude Current = charges in motion Magnitude rate at which net positive charges move across a cross sectional surface J = current density (vector) in A/m² Units: [I] = C/s = A (ampere) Current is a scalar, signed quantity, whose sign corresponds to the direction of motion of net positive charges by convention

Ohm’s Law and non linear devices Current-Potential (I-V) characteristic of a device may or may not obey Ohm’s Law: or V = IR with R constant Resistance (ohms) gas in fluorescent tube tungsten wire diode

Energy in Electric Circuits Steady current means a constant amount of charge ΔQ flows past any given cross section during time Δt, where I= ΔQ / Δt. Energy lost by ΔQ is V => heat Read 27-8 (semi-conductor) and 27-9 (superconductor) So, Power dissipation = rate of decrease of U =

Resistors in Parallel Devices in parallel has the same potential drop Generally, •••

Kirchhoff’s Rules Kirchhoff’s Rule 1: Loop Rule When any closed loop is traversed completely in a circuit, the algebraic sum of the changes in potential is equal to zero. Coulomb force is conservative Kirchhoff’s Rule 2: Junction Rule The sum of currents entering any junction in a circuit is equal to the sum of currents leaving that junction. Conservation of charge In and Out branches Assign Ii to each branch

Loop current example (with parallel R combos) Sketch the diagram Simplify using equivalent resistors Label loop currents with directions Use Loop currents I1 and I2 Choose interior clockwise loops . Set up cononical equations in . I1 I2 Ε format I1 -I2 Replace by equivalent R=2Ω first. I1 (12 +6) +I2 (-6) = +18 (Emf) Left loop I1 (-6) + I2 (6+3+2) =+21 (Emf) Right loop Note Symmetry of Equations

Galvanometer Inside Ammeter and Voltmeter Galvanometer: a device that detects small currents and indicates its magnitude. Its own resistance Rg is small for not disturbing what is being measured. shunt resistor galvanometer Ammeter: an instrument used to measure currents Voltmeter: an instrument used to measure potential differences galvanometer

Discharging a Capacitor in RC Circuits Switch closed at t=0. Initially C is fully charged with Q0 Loop Rule: I Convert to a differential equation Solve it!

Charging a Capacitor in RC Circuits Switch closed at t=0 C initially uncharged, thus zero voltage across C. Loop Rule: 3. Convert to a differential equation Solve it! (τ=RC is the time constant again)