Chapter 23: Gauss Law. PHY2049: Chapter 23 1

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1 Chapter 23: Gauss Law PHY2049: Chapter 23 1

2 Two Equivalent Laws for Electricity Coulomb s Law equivalent Gauss Law Derivation given in Sec (Read!) Not derived in this book (Requires vector calculus) We will focus on what is Gauss law and how we use it. PHY2049: Chapter 23 2

3 Electric Flux Simple definition of electric flux (E constant, flat surface) E at an angle θ to planar surface, area A ΦE E A= EAcosθ Units = N m 2 / C (SI units) Normal E Simple example Let E = 10 4 N/C pass through 2m x 5m rectangle, 30 to normal φ E = 10 4 * 10 * cos(30 ) = 100,000 * = 86,600 More general Φ E definition (E variable, curved surface) ΦE E d A S PHY2049: Chapter 23 3

4 Example of Constant Field E = 4iˆ N/C A = ( 2ˆ i + 3 ˆ) j 2 m Φ E E A = 4ˆ i (2ˆ i + 3 ˆ) j = 8 Nm 2 / C PHY2049: Chapter 23 4

5 Flux Through Closed Surface Surface elements da always point outward! Sign of Φ E E outward (+) E inward ( ) Φ E < 0 Φ E > 0 Φ E = 0 PHY2049: Chapter 23 5

6 Example: Flux Through Cube E field is constant: r E = Ezˆ Flux through front face? Flux through back face? Flux through top face? Flux through whole cube? PHY2049: Chapter 23 6

7 Example: Flux Through Cylinder Assume E is constant, to the right Flux through left face? Flux through right face? Flux through curved side Total flux through cylinder? PHY2049: Chapter 23 7

8 Example: Flux Through Sphere Assume point charge +Q E points radially outward (normal to surface!) Φ E = E d A S kq = 2 r = 4π kq = ( 2 4π r ) Q ε 0 Foreshadowing of Gauss Law! PHY2049: Chapter 23 8

9 Gauss Law General statement of Gauss law S E da = q enc Can be used to calculate E fields. But remember Outward E field, flux > 0 Inward E field, flux < 0 ε 0 Integration over closed surface q enc is charge inside the surface Charges outside surface have no effect (This does not mean they do not contribute to E.) Consequences of Gauss law (as we shall see) Excess charge on conductor is always on surface E is always normal to surface on conductor (Excess charge distributes on surface in such a way) Conductor PHY2049: Chapter 23 9

10 Reading Quiz What is the electric flux through a sphere of radius R surrounding a charge +Q at the center? 1) 0 2) +Q/ε 0 3) Q/ε 0 4) +Q 5) None of these PHY2049: Chapter 23 10

11 ds Question ds +Q +Q 1 2 How does the flux Ф E through the entire surface change when the charge +Q is moved from position 1 to position 2? a) Ф increases E b) Ф decreases E c) Ф E doesn t change Just depends on charge not position PHY2049: Chapter 23 11

12 Power of Gauss Law: Calculating E Fields Valuable for cases with high symmetry E = constant, surface E surface E da = ± EA Spherical symmetry E field vs r for point charge E field vs r inside uniformly charged sphere Charges on concentric spherical conducting shells Cylindrical symmetry E field vs r for line charge E field vs r inside uniformly charged cylinder Rectangular symmetry E field for charged plane E field between conductors, e.g. capacitors S S E da = 0 PHY2049: Chapter 23 12

13 Example 4 Gaussian surfaces: 2 cubes and 2 spheres Rank magnitudes of E field on surfaces Which ones have variable E fields? What are the fluxes over each of the Gaussian surfaces (a) E falls as radius increases (b) E non-constant on cube (r changes) (c) Fluxes are same, +Q/ε 0 PHY2049: Chapter 23 13

14 Derive Coulomb s Law From Gauss Law Charge +Q at a point By symmetry, E must be radially symmetric Draw Gaussian surface around point Sphere of radius r E field has constant mag., to Gaussian surface r E S da = E(4πr 2 ) = Q ε 0 Gauss Law Gaussian surface (sphere) E Q = = 4πε r 0 kq 2 2 r Solve for E PHY2049: Chapter 23 14

15 Example Charges on shells are +Q (ball at center) +3Q (middle shell) +5Q (outside shell) Find fluxes on the three Gaussian surfaces (a) Inner +Q/ε 0 (b) Middle +4Q/ε 0 (c) Outer +9Q/ε 0 PHY2049: Chapter 23 15

16 Conductors with No Current E is zero everywhere inside Why? Conductors are full of mobile charges (e.g., conduction electrons in a background formed by immobile positive ions). If there were E, then the charges must be moving around due to force F=qE. This would contradict no current. Note: even if there is an externally imposed E, it cannot go inside All excess charge must be on outer surface. Why? Since E=0 everywhere inside, q enc enclosed by any Gaussian surface is also zero everywhere inside. Note: distribution of surface charge must be such to make E=0 everywhere inside E is always normal to surface on conductor Why? E component parallel to surface would cause surface charge to move. This would contradict no current. PHY2049: Chapter 23 16

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