Physics 2113 Physics 2113 Lecture: 09 WED 04 FEBLecture: 09 WED 04 FEB

GaussGauss’’ Law I Law I

Flux Capacitor (Schematic)

Physics 2113

Jonathan Dowling

Carl Friedrich Gauss1777 – 1855

What Are We Going to Learn?What Are We Going to Learn?A Road MapA 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!What? — The Flux!STRONGE-Field

WeakE-Field

Number of E-Lines Through Differential

Area “dA” is a Measure of Strength

dAθ

Angle Matters Too

Electric Field & Force Law Electric Field & Force Law Depends on GeometryDepends 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 SurfaceElectric 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 CaseElectric Flux: The General Case

Air Flow Analogy

Electric Flux: ICPPElectric Flux: ICPP

• 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

Hint!Surface area of sides of cylinder: 2πRLSurface area of top and bottom caps (each): πR2

L

R

E

(πR2)E–(πR2)E=0What goes in — MUST come out!

dA

dA

(a) +EA? –EA? 0?

(b) +EA? –EA? 0?

(c) +EA? –EA? 0?

(c) +EA? –EA? 0?

Electric Flux: ICPPElectric 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?

(a) (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?

q

r

Electric Flux: ExampleElectric Flux: Example

Since r is Constant on the Sphere — RemoveE Outside the Integral!

Gauss’ Law:Special Case!

(Outward!)

Surface Area Sphere

(Inward!)

q

r

Gauss’s Law: Gravitational Field vs Electric FieldGauss’s Law: Gravitational Field vs Electric Field

r

M

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