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Unformatted text preview: 1 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Lecture 9 Magnetoquasistatics In this lecture you will learn: • Basic Equations of Magnetoquasistatics • The Vector Potential • The Vector Poisson’s Equation • The BiotSavart Law • Magnetic Field of Some Simple Current Carrying Elements • The Magnetic Current Dipole ECE 303 – Fall 2007 – Farhan Rana – Cornell University Equations of Magnetoquasistatics ( ) t r E o , . r r ρ ε = ∇ = × ∇ E r . = ∇ H o r µ ( ) t r J H , r r r = × ∇ Equations of Electroquasistatics Equations of Magnetoquasistatics • Electric fields are produced by only electric charges • Once the electric field is determined, the magnetic field can be found by the last equation above • Magnetic fields are produced by only electric currents • Once the magnetic field is determined, the electric field can be found by the last equation above • Currents in magnetoquasistatics are solenoidal (i.e. with zero divergence) t E J H o ∂ ∂ + = × ∇ r r r ε t H E o ∂ ∂ − = × ∇ r r µ In magnetoquasistatics the source of the magnetic field is electrical current ( ) ( ) . , . = × ∇ ∇ = ∇ H t r J r r r 2 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Ampere’s Law for Magnetoquasistatics ∫∫ = ∫ a d J s d H r r r r . . A closed contour Ampere’s Law : The line integral of magnetic field over a closed contour is equal to the total current flowing through that contour Right Hand Rule: The positive directions for the surface normal vector and of the contour are related by the right hand rule electric current density J H r r = × ∇ ECE 303 – Fall 2007 – Farhan Rana – Cornell University Magnetic Field of an Infinite LineCurrent Consider an infinitely long linecurrent carrying a total current I in the + zdirection, as shown below x y line current Use ampere’s law on the closed contour shown by the dashed line: ( ) ( ) ( ) r I r H I r H r a d J s d H π π φ φ 2 2 . . = ⇒ = ⇒ ∫∫ = ∫ r r r r r Magnetic field is entirely in the direction and falls off as ~1/ r from the linecurrent φ ˆ Working in the cylindrical coordinates s d r 3 ECE 303 – Fall 2007 – Farhan Rana – Cornell University Magnetic Field of a Solenoid Consider a solenoid with N turns per unit length and carrying a current I • The magnetic field inside the solenoid is uniform and strong • There is a fringing field outside the solenoid which is very weak and may be neglected Assumptions: I H L ( ) NI H I LN H L a d J s d H y y = ⇒ = ⇒ ∫∫ = ∫ r r r r . ....
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 Fall '06
 RANA
 Electromagnet

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