1. Flux Capacitor (Schematic)
Physics 2113
Jonathan Dowling
Flux Capacitor (Schematic)
Physics Lecture: 08
Gauss’ Law I
Carl Friedrich Gauss
1777 – 1855

2. 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)

3. 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

4. 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!

5. What? The 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

7. What? The Flux! General Case
Air Flow Analogy

8. What? The Flux! General Surface
For any general surface: break up into infinitesimal planar patches
Electric Flux
Surface integral
dA is a vector normal to each patch and has a magnitude |dA|= dA
CLOSED surfaces:
define the vector dA as pointing OUTWARDS
Inward E gives negative flux 𝚽
Outward E gives positive flux 𝚽 E dA dA E
Area = dA

9. What? The 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

11. (a) front
+EA?
–EA? 0?
(b) rear
+EA?
–EA? 0?
(c) top
+EA?
–EA? 0?
(c) Whole cube
+EA?
–EA? 0?

12. 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 = π 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?

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

14. Gauss’s Law: Gravitational Field vs Electric FieldM r q

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