Ex. 2: Integration - Disk of charge r k(dq)z ˆ k(σ 2πrdr)z ˆ Circular filament dE = 2 2 3/ 2 k = 2 2 3/ 2 k (we already know (z + r ) (z + r ) the field at P from it!) r ˆ a 2rdr Field is along k E = kσπzk∫ r=0 (z2 + r2)3/ 2 σ=Q/(πa2) v  z  E = 2kσπkˆ1−  Integrate over set  2 2  of rings, each of  z + a  radius r and thickness dr, with 2kQ  z  charge dQ= = kˆ1−  σ2πrdr 2   a  z2 + a2 

Volume charge density ρ and Gauss’s law h • Volume charge density ρ : dQ=ρdV; z w z • If volume charge density is uniform, ρ=Q/V • Example: Infinite “slab” of charge (Gauss’s law) h

Gaussian Uniform volume charge density ρ surface for outside field Field above slab same as that of infinite sheet of charge with σ = ρw r ρw ρ E = kˆ 2ε0 Gaussian Inside slab, use Gaussian box going from -z to +z with surface for area A top and bottom. Again, E = E k, so inside field

Q in ρ (2 z ) A Motion of charge in Electric Field Φ E = EA + EA = = ε 0 ε 0 • Nothing new - electric force acts like any other force; 2 ρzA charge q of mass m in field E has acceleration: 2 EA = q r ε ar = E 0 m ρz E = • Object containing equal amounts of pos. & neg. ε 0 charge will not feel force in uniform E, but will feel r ρz w E = kˆ.... for z < force if E varies with position. ε 0 2 Note that the sign automatically changes for z<0 Question: Start at midpoint and move up; how does E change?

1 Electric Dipole Forces and Torques on Dipole • Common arrangement of charges: • Dipole placed in uniform field feels no net force -Q +Q r r r Ftotal = QE + (−Q)E = 0 - + • Feels net torque : r r r τtotal = ∑ri × Fi l i r r r r1 = 0....r2 = l r r r r r p τtotal = l ×QE = Ql × E • No net charge on dipole, but charges separated Thus torque on dipole is • Example: water molecule r r r • Dipole moment p = Q l where l is the vector from τ = p × E the negative charge to the positive charge Question: Direction of torque?

Torque example Electric Potential Energy •Remember gravitational potential energy U = mgh • What is the magnitude of the torque on a dipole for object of mass m at height h above zero potential with p = 0.01 C•m and in field of strength E = 100 reference position (valid near Earth surface) N/C if the angle between p and E is 30°? – Describes energy stored when work done against • Is the torque on a dipole in an E field ever zero? 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) Coupling strength Field Potential Energy Mass Gravitational Gravitational Charge Electric Electric

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