MAGNETIC FIELDS DUE TO CURRENTS

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PHY110W MAGNETISM MAGNETIC FIELDS DUE TO CURRENTS AMPERE’S LAW Gauss’s law related the surface integral of the electric flux over a closed (Gaussian) surface to the net amount of charge enclosed by that surface. Ampere’s law relates the line integral of the magnetic field around a closed (Amperian) loop to the net amount of current enclosed by that loop. Mathematically: ∫ B ⋅ ds = µ 0iencl (30-16) Amperian loop i 3 i 1 i 2 ds θ B According to the right hand curled fingers rule, as we integrate anticlockwise around the above loop i 1 is regarded as positive, while i 2 is negative. i 3 is ignored because it is not enclosed by the loop. B⋅ ds = Bdscosθ = µ i − i ∫ ∫ 0 1 2 (30-18) So: ( ) In the same way Gauss’s law helped to determine the magnitude of electric fields due to symmetrical charge distributions, Ampere’s law is useful in determining the magnitude of the resultant magnetic fields due to current distributions which have some symmetry… 20
PHY110W MAGNETISM MAGNETIC FIELDS DUE TO CURRENTS APPLICATIONS OF AMPERE’S LAW I Magnetic field outside a long, straight conductor For a long, straight wire carrying current directly out of the page the symmetry dictates that we surround the wire with a concentric circular Amperian loop of radius r. Amperian loop i ds B r B lies at a tangent to the loop everywhere on the loop, so the angle θ between B and ds is zero. ∴ B⋅ ds = B ds = B2π r ∫ ∫ And applying Ampere’s law: B2π r = µ ( ) 0 + i So the magnitude of the magnetic field around a long, straight, current-carrying conductor is B µ i 2π r 0 = (30-19) (Which agrees exactly with what was derived from the Biot-Savart law.) 21
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MAGNETIC FIELDS DUE TO CURRENTS

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