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Ex. 2: Integration - Disk of charge Circular filament (we already know the field at P from it!) Field is along k σ=Q/(πa2) Integrate over set of rings, each of radius r and thickness dr, with charge dQ= σ2πrdr r dE = k (dq ) z ˆ k (σ 2πrdr ) z ˆ k k= 2 ( z 2 + r 2 )3 / 2 ( z + r 2 )3 / 2 r 2rdr a E = kσπzkˆ ∫r = 0 2 ( z + r 2 )3 / 2 v z E = 2kσπkˆ1 − 2 z + a2 = 2kQ ˆ z k 1 − 2 a z 2 + a2 Volume charge density ρ and Gauss’s law h • Volume charge density ρ : dQ=ρdV; • If volume charge density is uniform, ρ=Q/V • Example: Infinite “slab” of charge (Gauss’s law) z z Gaussian surface for outside field ρ Gaussian surface for inside field Φ E = EA + EA = 2 EA = E = Q in ρ (2 z ) A = ε0 ε0 2 ρ zA ε0 ρz ε0 r w ρz ˆ E = k .... for z < 2 ε0 w h Uniform volume charge density ρ Field above slab same as that of infinite sheet of charge with σ = ρw r ρw E= 2ε 0 kˆ Inside slab, use Gaussian box going from -z to +z with area A top and bottom. Again, E = E k, so Motion of charge in Electric Field • Nothing new - electric force acts like any other force; charge q of mass m in field E has acceleration: r q r a = E m • Object containing equal amounts of pos. & neg. charge will not feel force in uniform E, but will feel force if E varies with position. Note that the sign automatically changes for z<0 Question: Start at midpoint and move up; how does E change? 1 Electric Dipole • Common arrangement of charges: Forces and Torques on Dipole -Q +Q • Dipole r placed in r uniform rfield feels no net force - + • Feels net torque : l p • No net charge on dipole, but charges separated • Example: water molecule • Dipole moment p = Q l where l is the vector from the negative charge to the positive charge Torque example • What is the magnitude of the torque on a dipole with p = 0.01 C•m and in field of strength E = 100 N/C if the angle between p and E is 30°? • Is the torque on a dipole in an E field ever zero? Ftotal = QE + (−Q ) E = 0 r r r τ total = ∑ ri × Fi i r r r r1 = 0....r2 = l r r r r r τ total = l × QE = Ql × E r r r τ = p×E Thus torque on dipole is Question: Direction of torque? Electric Potential Energy • Remember gravitational potential energy U = mgh for object of mass m at height h above zero potential reference position (valid near Earth surface) – Describes energy stored when work done against conservative force (gravity) – g is strength of Earth gravitational field (in N/kg) – We must do work on mass m to increase its gravitational potential energy • A charged particle in an electric field has electric potential energy (electric force is conservative) Potential Energy Coupling strength Field Mass Gravitational Gravitational Charge Electric Electric 2