EGR-326 ELECTROMAGNETIC FIELDS NO FLUX GIVEN SEPTEMBER 8, 2023 DORDT UNIVERSITY FALL 2023 DR. WYENBERG STREAMLINES -1 +1 -1 +2 -1 CHAPTER 3 – ELECTRIC FIELD DENSITY • Geometrical interpretation of Electric Fields – introduction of Electric Flux Ψ = ππ π·π· = ππ0 πΈπΈ = ππ ππΜ 4ππππ 2 π·π· is measured in Coulombs per square meter (remember that ππ0 is the conversion factor between electric charge and force; it has units of Coulombs squared per Newton meter squared) GAUSS’ LAW οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ Use Gauss’ Law to calculate the flux density (and hence the electric field strength): 1. Surrounding a point charge ππ 2. Surrounding a line charge with charge density ππππ 3. Surrounding a plane with charge density πππ π οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ Use Gauss’ Law to calculate the charge enclosed in a small volume of space οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ ππππππππ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + βππ = ππππ πππ¦π¦ πππ§π§ ππππ = πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + ππππ πππ¦π¦ πππ§π§ π₯π₯, π¦π¦, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯ π§π§ +1C −1C π¦π¦ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π·π·π§π§ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ π₯π₯, π¦π¦, π§π§ + βππ π₯π₯ + βππ, π¦π¦, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ + βππ π·π·π§π§ π₯π₯, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ + βππ π₯π₯, π¦π¦ + βππ, π§π§ π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯ + βππ, π¦π¦ + βππ, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = π·π·π₯π₯ π₯π₯ + βππ π΄π΄ οΏ½ π·π·ππ οΏ½ ππ ππβ = −π·π·π₯π₯ π₯π₯ π΄π΄ ππ ππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππ ππβ = π·π·π₯π₯ π₯π₯ + βππ βππ2 οΏ½ π·π·ππ οΏ½ ππ ππβ = −π·π·π₯π₯ π₯π₯ βππ2 ππ ππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππ ππβ = π·π·π₯π₯ π₯π₯ + βππ βππ 2 − π·π·π₯π₯ π₯π₯ βππ2 ππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π₯π₯ + βππ, π¦π¦, π§π§ π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππ ππβ = π·π·π₯π₯ π₯π₯ + βππ βππ 2 − π·π·π₯π₯ π₯π₯ βππ2 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π·π·π₯π₯ π₯π₯ + πΌπΌ ≈ π·π·π₯π₯ π₯π₯ + πΌπΌ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππ ππβ = π·π·π₯π₯ π₯π₯ + βππ βππ 2 − π·π·π₯π₯ π₯π₯ βππ2 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = π·π·π₯π₯ π₯π₯ + βππ ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ βππ 2 − π·π·π₯π₯ π₯π₯ βππ2 πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = βππ 3 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = βππ 3 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ π₯π₯, π¦π¦, π§π§ ππππ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = βππ 3 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ 6C/M 2 3 = βππ ππππ βππ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππππβ = βππ 3 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + ππππ πππ¦π¦ πππ§π§ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ οΏ½ π·π·ππ οΏ½ ππ ππβ = βππ 3 ππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + = ππππππππ ππππ πππ¦π¦ πππ§π§ π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ ππππππππ + + = ππππ ππππ ππππ βππ 3 π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ ππππππππ + + = ππππ ππππ ππππ π£π£π£π£π£π£ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + = πππ£π£ ππππ ππππ ππππ π·π·π₯π₯ π₯π₯ + βππ = −1C/m2 π·π·π₯π₯ π₯π₯ + βππ ≈ π·π·π₯π₯ π₯π₯ + βππ π₯π₯ + βππ, π¦π¦, π§π§ πππ·π·π₯π₯ π₯π₯ ππππ π·π·π₯π₯ π₯π₯ = −7.5C/m2 π₯π₯, π¦π¦, π§π§ Implies… οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + ππππ = ππππ ππππ ππππ ππππ = πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + ππππ ππππ ππππ ππ ππ ππ ππππ = π₯π₯οΏ½ + π¦π¦οΏ½ + π§π§Μ οΏ½ π·π· ππππ ππππ ππππ ππ ππ ππ ∇= π₯π₯οΏ½ + π¦π¦οΏ½ + π§π§Μ → ππππ = ∇ οΏ½ π·π· ππππ ππππ ππππ ∇ is the “del” operator, and ∇ οΏ½ π·π· is the “divergence of π·π·.” “The charge density in space is the divergence of the electric flux density.” Which should make sense, because electric flux is created by charge! DIVERGENCE OF OTHER VECTOR FIELDS • Divergence of water in a bathtub vs. air in a popped tire DIVERGENCE THEOREM ππππ = ∇ οΏ½ π·π· οΏ½ π·π·ππ οΏ½ ππππβ = οΏ½ πππ£π£ ππππ ππ π£π£ππππ οΏ½ π·π·ππ οΏ½ ππ ππβ = οΏ½ ∇ οΏ½ π·π·ππππ ππ π£π£ππππ DEL OPERATOR IN OTHER COORDINATE SYSTEMS ∇= ππ ππ ππ πππ·π·π₯π₯ πππ·π·π¦π¦ πππ·π·π§π§ + + π₯π₯οΏ½ + π¦π¦οΏ½ + π§π§Μ → ∇ οΏ½ π·π· = ππππ ππππ ππππ ππππ ππππ ππππ 1 ππ 1 ππ ππ 1 ππ 1 πππ·π·ππ πππ·π·π§π§ οΏ½ + ∇= ππβ + (πππ·π·ππ ) + ππ + π§π§Μ → ∇ οΏ½ π·π· = ππππ ππ ππππ ππ ππππ ππππ ππ ππππ ππ ππππ 1 ππ 1 ππ 1 ππ 1 ππ 2 1 ππ 1 πππ·π·ππ Μ οΏ½ ∇= 2 ππππβ + (sin ππ ππ) + ππ π·π·ππ + (sin ππ π·π·ππ ) + ππ → ∇ οΏ½ π·π· = 2 ππ ππππ ππ sin ππ ππππ ππ sin ππ ππππ ππ ππππ ππ sin ππ ππππ ππ sin ππ ππππ DEL OPERATOR IN OTHER COORDINATE SYSTEMS 1 ππ 1 πππ·π·ππ πππ·π·π§π§ + ∇ οΏ½ π·π· = πππ·π·ππ + ππππ ππ ππππ ππ ππππ π·π·ππ πππ·π·ππ 1 πππ·π·ππ πππ·π·π§π§ + + + ∇ οΏ½ π·π· = ππ ππππ ππ ππππ ππππ DEL OPERATOR IN OTHER COORDINATE SYSTEMS 1 ππ 2 1 ππ 1 πππ·π·ππ ∇ οΏ½ π·π· = 2 ππ π·π·ππ + (sin ππ π·π·ππ ) + ππ ππππ ππ sin ππ ππππ ππ sin ππ ππππ 2π·π·ππ πππ·π·ππ 1 πππ·π·ππ 1 πππ·π·ππ + + + ∇ οΏ½ π·π· = cot ππ π·π·ππ + ππ ππππ ππππ ππ ππ sin ππ ππππ DIVERGENCE THEOREM EXAMPLE Check that the divergence theorem holds for the electric flux density caused by a 1C charge at the origin: 1Cππβ π·π· = 4ππππ 3 DIVERGENCE THEOREM EXAMPLE π·π· = 1Cππβ 4ππππ 3 οΏ½ π·π·ππ οΏ½ ππ ππβ = οΏ½ ∇ οΏ½ π·π·ππππ ππ π£π£ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 1Cππβ 1Cππβ οΏ½ ππ Μ = οΏ½ ∇ οΏ½ ππππ 4ππππ 3 4ππππ 3 π£π£ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 ππ ππβ = οΏ½ ∇ οΏ½ ππππ ππ 3 ππ 3 π£π£ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 1 1 ππ 1 ππ 1 ππ ππβ Μ + οΏ½ οΏ½ ππππ = οΏ½ ππ ππ β + sin ππ ππ ππ ππ 2 ππ 2 ππππ ππ sin ππ ππππ ππ sin ππ ππππ ππ 3 π£π£ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 1 1 ππ ππβ = οΏ½ ππ ππ β οΏ½ ππππ ππ 2 ππ 2 ππππ ππ 3 π£π£ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 1 1 ππ ππβ = οΏ½ ππ ππ β οΏ½ ππ 2 ππ 2 ππππ ππ 3 π£π£ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = οΏ½ ππ ππβ 4ππππ 3 1 1 ππ = οΏ½ 1 ππ 2 ππ 2 ππππ π£π£ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = ππ 2ππ οΏ½οΏ½ 0 0 ππβ 4ππππ 3 1 1 ππ ππ sin ππ ππππππππππ = οΏ½ 1 ππ 2 ππ 2 ππππ π£π£ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = ππ 2ππ ππβ 4ππππ 3 οΏ½ οΏ½ sin ππ ππππππππ = οΏ½ 0 0 π£π£ππππ 1 ππ 1 ππ 2 ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = ππ ππβ 4ππππ 3 2ππ οΏ½ sin ππ ππππ = οΏ½ 0 π£π£ππππ 1 ππ 1 ππ 2 ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = 4ππ = οΏ½ π£π£ππππ ππβ 4ππππ 3 1 ππ 1 ππ 2 ππππ ππππ DIVERGENCE THEOREM EXAMPLE π·π· = π π ππ 2ππ ππβ 4ππππ 3 1 ππ 4ππ = οΏ½ οΏ½ οΏ½ 2 1 ππ 2 sin ππ ππππππππππππ ππ ππππ 0 0 0 DIVERGENCE THEOREM EXAMPLE ππβ π·π· = 4ππππ 3 ππ 4ππ = 4ππ οΏ½ 0 ππ 1 ππππ ππππ DIVERGENCE THEOREM EXAMPLE ππβ π·π· = 4ππππ 3 ππ 4ππ = 4ππ οΏ½ 0 ππ 1 ππππ ππππ