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PHY2054 Fall 2015 PHY2054 Exam 1 Formula Sheet Vectors r r r r a = a x xˆ + a y yˆ + a z zˆ b = bx xˆ + by yˆ + bz zˆ Magnitudes: a = a x2 + a y2 + a z2 b = bx2 + by2 + bz2 r r r r r r Scalar Product: a ⋅ b = a xbx + a y by + a z bz = a b cos θ (θ = angle between a and b ) Electrostatic Force and Electric Field r qq Electrostatic Force (vector): F = k 1 2 2 rˆ (r = distance between charge q1 and charge q2, units = N) r ε0 = 8.85x10-12 C2/(N·m2) k = 1/(4πε0) = 8.99x109 N·m2/C2 r r q E = F / q2 = k 12 rˆ (units = N/C = V/m) r r r Electric Flux (through the infinitesimal surface area dA): dΦ E = E ⋅ dA (units = Nm2/C) r Vector Area (directed area): A = Anˆ (where n̂ = normal to the surface) r r Q Gauss’ Law (net flux through closed surface S): Φ E = ∫ E ⋅ dA = enclosed (Qenclosed = charge enclosed) Electric Field (at q2 due to q1): ε0 S Gauss’ Law: Net flux through closed surface S = the charge enclosed by surface S divided by ε0 Electric Potential and Potential Energy r r straight line path from A to B, ΔU = U B − U A = − qE ⋅ d (units = J) Electric Potential Energy: work done against a constant field E in moving charge q a distance d along Electric Potential: Work done per unit charge against a constant field E in moving charge q a distance d r r along straight line path from A to B, ΔV = ΔU / q = − E ⋅ d (units = J/C = V) Electric Potential (distance r from a point charge q): V (r ) = k Electric Potential Energy (N point charges): U = N q q N point charges: V (r ) = ∑ k i r ri i =1 N 1 2 V is the electric potential at q due ∑ q V , where to the other charges i =1 i i i i Stored Electric Potential Energy (N conductors with charge Qi and electric potential Vi): U= N 1 2 ∑QV i =1 i i Capacitance (definition): C = Q/V or C = Q/ΔV (units = C/V = F) Energy Density of the Electric Field: u = 12 ε 0 E 2 (units = J/m3) Electric Current and Circuits Current (through directed area A): I = r r dQ r r = J ⋅ A = nqvdrift ⋅ A (units = C/s = A, n is the number of dt charged particles q per unit volume, vdrift is the average velocity of the charged particles). Conducting Wire (length L, cross sectional area A): J = σE , Ohm’s Law: ΔV = IR , I = σEA = EA / ρ , ΔV = EL R = ρL / A (Resistance R units = V/A = Ω) Resistivity (at temperature T in oC): ρ (T ) = ρ 0 (1 + αΔT ) , where ΔT = T − T0 Power (supplied by EMF ε): P = εI Power (dissipated in resistor R): P = I2R (units = J/s = W) RC Circuits (charging capacitor C through resistor R with EMF ε): Q(t ) = εC (1 − e −t / τ ) RC Circuits (discharging capacitor C with initial charge Q0 through resistor R): Q(t ) = Q0e − t / τ RC Circuits (time constant): τ = RC (units = Ω·F = s) Page 1 of 1