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# c1 - Maxwell's Equations Gauss Faraday Ampere E = 0 H = 0 E...

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Maxwell’s Equations Gauss ∇ · εE = ρ εE · da = ρdV ∇ · μ 0 H = 0 μ 0 H · da = 0 Faraday ∇ × E = - ∂μ 0 H ∂t E · ds = - ∂t μ 0 H · da Ampere ∇ × H = J + ∂εE ∂t H · ds = J · da + ∂t εE · da Boundary Conditions D 2 - D 1 = σ u E 2 - E 1 = 0 ε 0 ( E 2 - E 1 ) = σ u + σ p B 2 - B 1 = 0 H 2 - H 1 = 0 Common Electric Fields Point Charge E r = q 4 πε 0 r 2 Infinite Charge Plane E x = σ 2 ε 0 Uniformly Charged Sphere E r = ρ 3 ε 0 r, r < a E r = ( ρ 4 3 πa 3 ) 4 πε 0 r 2 , r > a Charge Ring E z = λ 2 πa 4 πε 0 z 2 Line Charge E z = λL 4 πε 0 z 2 Oppositely Charged Plates E = σ ε 0 Charge Dipole φ ( r ) = qd 4 πε 0 r 3 (2 cos θ ˆ r + sin θ ˆ θ ) Electric Potential φ ( r ) = r E · ds E = -∇ φ ( r ) 2 φ = ρ ε 0 Point Charge φ ( r ) = q 4 πε 0 r Charge Dipole φ ( r ) = qd 4 πε 0 r 2 cos θ ↓↑ Charge Quadrupole φ ( r ) = qd 2 4 πε 0 r 3 sin(2 θ ) Charge Dipole p = qd P = pN = ε 0 χ e E ε = ε 0 (1 + χ e ) φ ( r ) = p cos θ 4 πε 0 r 2 Capacitance C = dQ dV = Q V Laplacian Trial Solutions (Spherical) Constant φ ( r ) = A Spherically Symm. φ ( r ) = A r + B Uniform z-dir.

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c1 - Maxwell's Equations Gauss Faraday Ampere E = 0 H = 0 E...

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