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Ampere’s Law. Alan Murray. dl. B. B. I. B. dl. dl. B. Try this …. Create a contour for integration (a circle seems to make sense here!). Take a closed contour. These currents are “enclosed”. And these currents are not!. ò H . dl = Current I enclosed.

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## Ampere’s Law

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**Ampere’s Law**Alan Murray**dl**B B I B dl dl B Try this … Create a contour for integration (a circle seems to make sense here!) Alan Murray – University of Edinburgh**Take a closed contour**These currents are “enclosed” And these currents are not! òH.dl = Current Ienclosed • This is, as it turns out, Ampere’s Law and is the magnetic-field equivalent of Gauss’s law • If we define H=B÷μ, B=μH, thenòH.dl = Current “enclosed” = òòòJ.ds I4 I5 I1 I3 I2 I6 Alan Murray – University of Edinburgh**òH.dl = Current Ienclosed**I4 I1 I3 I2 Alan Murray – University of Edinburgh**|J| = I/A**B? I B? R Ampere’s Law – Worked Example • Calculate the magnetic field H both Outside (r>R) and Inside (r<R) A wire with uniformly-distributed current I,current density Alan Murray – University of Edinburgh**I, |J|=I/pR2**H, B R Outside … r>R, òH.dl = Ienclosed B r Alan Murray – University of Edinburgh**I, |J|=I/pR2**I =I pr2pR2 Inside … r<R, òH.dl = Ienclosed B? R r Alan Murray – University of Edinburgh**r**B,H Inside … r<R, òH.dl = Ienclosed Alan Murray – University of Edinburgh**L**B-field here? I Ampere – Worked example i.e. what is the magnetic fieldabove and close to a metal “track”on a printed circuit board or chip? Alan Murray – University of Edinburgh**I**What does the B-field look like? B-field lines Alan Murray – University of Edinburgh**Width L**Contour for integration b a Current enclosed = a´I/L I/L Amps/metre of width, out of the diagram òH.dl = Ienclosed? Alan Murray – University of Edinburgh**òH.dl = aH(H, dl parallel)**òH.dl = 0(H, dl perpendicular) òH.dl = 0(H, dl perpendicular) H òH.dl = aH(H, dl parallel) òH.dl = Ienclosed? H a b b a Current enclosed = a´I/L òH.dl = aH + aH = 2aH òH.dl = Ienclosed® 2aH = aI/L |H| = _I_ 2L Alan Murray – University of Edinburgh**òH.dl = Ienclosed?**|H| = -I_ 2L |H| = +I_ 2L Alan Murray – University of Edinburgh

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