Physics 2113 Lecture: 09 WED 04 FEB Jonathan Dowling Flux Capacitor (Schematic) Physics 2113 Lecture: 09 WED 04 FEB Gauss’ Law I Carl Friedrich Gauss 1777 – 1855

What Are We Going to Learn? A Road Map Electric charge - Electric force on other electric charges - Electric field, and electric potential Moving electric charges : current Electronic circuit components: batteries, resistors, capacitors Electric currents - Magnetic field - Magnetic force on moving charges Time-varying magnetic field - Electric Field More circuit components: inductors. Electromagnetic waves - light waves Geometrical Optics (light rays). Physical optics (light waves)

What? — The Flux! STRONG E-Field Angle Matters Too Weak E-Field θ dA Number of E-Lines Through Differential Area “dA” is a Measure of Strength dA

Electric Field & Force Law Depends on Geometry Point of Charge: Field Spreads in 3D Like Inverse Area of Sphere = 1/(4πr2) Line of Charge: Field Spreads in 2D Like Inverse Circumference of Circle = 1/(2πr) Sheet of Charge: Field Spreads in 1D Like A Constant — Does Not Spread!

Electric Flux: Planar Surface Given: planar surface, area A uniform field E E makes angle θ with NORMAL to plane Electric Flux: Φ = E•A = E A cos  θ Units: Nm2/C Visualize: “Flow of Wind” Through “Window” θ E AREA = A=An normal

Electric Flux: The General Case Air Flow Analogy

Electric Flux: ICPP E dA Closed cylinder of length L, radius R (πR2)E–(πR2)E=0 What goes in — MUST come out! dA E Closed cylinder of length L, radius R Uniform E parallel to cylinder axis What is the total electric flux through surface of cylinder? (a) (2πRL)E (b) 2(πR2)E (c) Zero L R Hint! Surface area of sides of cylinder: 2πRL Surface area of top and bottom caps (each): πR2

(a) +EA? –EA? 0? (b) +EA? –EA? 0? (c) +EA? –EA? 0? (c) +EA? –EA? 0?

Electric Flux: ICPP Spherical surface of radius R=1m; E is RADIALLY INWARDS and has EQUAL magnitude of 10 N/C everywhere on surface What is the flux through the spherical surface? (4/3)πR3 E = -13.33π Nm3/C (b) 2πR E = -20π Nm/C (c) 4πR2 E= -40π Nm2/C What could produce such a field? What is the flux if the sphere is not centered on the charge?

Electric Flux: Example q r (Inward!) (Outward!) Since r is Constant on the Sphere — Remove E Outside the Integral! Surface Area Sphere Gauss’ Law: Special Case!

Gauss’s Law: Gravitational Field vs Electric Field M r q

ICPP: Compute the Surface Integral For each of the four Surfaces where + is a proton and – an electron