Formula Sheet

EP464 FORMULA SHEETPhysical Constants:Permittivity of free space: ε 0 = 8.85 × 10 -12 Farad/metrePermeability of free space: μ 0 = 4π × 10 -7 Tesla.metre/AmpereGauss’ Law∫ E• dS =Q inεEP464 Prof. M.P. Bradley Fall Term 2008Formula Sheet ρ∇ i E =ε0 ∂B∇× E =− ∂ tMaxwell’s Equations ∇ B = 0 Eμ ε ∂∂ti ∇× B = μ0J+0 0Definition of Scalar Potential, Poisson & Laplace EquationsPoisson’s Equation:Vb V − V =− E⋅d E = −∇Vb−ρεa∫a2∇ = for regions containing volume charge density ρ0Laplace’s Equation:2∇ V = 0 for charge-free regionsMagnetic Vector PotentialB= ∇× AVector Calculus Relations ∇× ( ∇ U ) = 0 ∇i( ∇× A) = 0 2∇ A=∇( ∇iA)−∇× ( ∇× A) ∇ ( UQ) = U ( ∇ Q) + Q( ∇U) ∇ i( UA) = U ( ∇ iA) + ( ∇U ) i A ∇ i( A× B) = ( ∇× A) iB− Ai( ∇× B) ∇× ( UA) = U ( ∇× A) + ( ∇ U ) × A 2∇i( ∇ U) =∇ U2 2 2 2∇ A= xˆ∇ A + yˆ∇ A + zˆ∇ Ax y z

Antenna TheoryEP464 Prof. M.P. Bradley Fall Term 2008Formula SheetElectric and Magnetic fields of a Hertzian dipole with drive current I0 and length Δ in the far-field: jβηI0Δ − jβRjβI0E = ˆ θ4πRsinθeH = ˆ φΔsinθe4πR− jβRμ0(Note that η ≡ is the impedance of free space for TEM wave propagation andε0the wavelength of electromagnetic waves in free space)2πβ = where λ isλTotal radiated power from a Hertzian Dipole:Prad , total2 2ηI0β=12π( Δ) 2Radiation Resistance of Hertzian Dipole:R rad2ηπ⎛Δ′⎞= ⎜ ⎟3 ⎝ λ ⎠2Definition of Antenna “directivity”: D( , )where:1 Pavg= Re ⎡E×H2 ⎣and P rad,total is the total radiated power*θφ ≡24 π RPavg( θφ , )Prad , total⎤⎦is the Radiation Poynting vector (power per unit area!)Antenna Gain , : For lossless antennas, max. Antenna Gain G = maximum directivity!Effective Aperture: , MORE FORMULAS ON NEXT PAGE

EP464 Prof. M.P. Bradley Fall Term 2008Formula SheetArbitrary-Length Dipole:Prad , totalηI=π ∫⎧ ⎡⎛βL⎞ ⎤ ⎛βL⎞⎫⎨cos ⎢⎜ ⎟cosθ− cos ⎬2⎥ ⎜ ⎟⎝ ⎠ ⎦ ⎝ 2 ⎠⎭θ2π0dθ⎩ ⎣40sin2The integral in the above formula (18.92 from the text) cannot be integrated analytically but is tabulatedand graphed below:Friis Transmision Formula:Radar Equation: 4