In electrostatics, the electric field E and its sources (charges) are related by Gauss s law: Surface

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1 Ampee s law n eletostatis, the eleti field E and its soues (hages) ae elated by Gauss s law: EdA i 4πQenl Sufae Why useful? When symmety applies, E an be easily omputed Similaly, in magnetism the magneti field and its soues (uents) ae elated by Ampee s law: 4π ds i enl Why useful? When symmety applies, E an be easily omputed N: This is a line integal! N: no demonstation has been given so fa fo Ampee s law. G. Siolla MT 8.0 etue Appliation of Ampee s law: eated by uent in a wie ong, staight wie in whih flows a uent alulate magneti field eated by Solution: Apply Ampee s law: 4 π ds i () π ˆ ϕ enl Dietion: ight hand ule N: wie ~ 1/. Does this look familia? Remembe E eated by a line of hage: oinidene? Not at all E () λ G. Siolla MT 8.0 etue

2 Foe between wies Foe on wie 1 due to magneti field eated by wie : F n ˆ 1 1 Magneti field eated by wie : ˆ ϕ 1 Total foe F: F F 1 Usually we quote the foe/unit length: Dietion? F 1 ϕˆ Using ight hand ule: 1 and paallel: attative 1 and anti-paallel: epulsive G. Siolla MT an we test this epeimentally? Demo G8, G9 8.0 etue Anothe appliation of Ampee s law: eated by sheet of uent alulate the magneti field eated by uent flowing in a sheet of onduto y uent // -z ais (into the page) Width of sheet of onduto: uent in a metal sheet ~ N paallel wies Solution: fom a wie is know: ˆ ϕ Just apply supeposition Dietion: fo y>0: // +; fo y<0: // - Magnitude: integate d field fom eah infinitesimal wie π ( ) When >>y, π / N: magnitude of does not depend on y. As fo E of sheet of hages G. Siolla MT 8.0 etue

3 Anothe appliation of Ampee s law: eated by plane of uent alulation: d (only omponent // ˆ suvives beause of symmety) / d / os d / os / / d os / yd os yd y os ytg d ; y os os os 4 ± ˆ + fo y>0; - fo y<0 y G. Siolla MT 8.0 etue Moe on fom sheet of uent f we define uent pe unit length K/: What is the hange of aoss the sheet of uent? 4 π K Does it ing a bell? Yes, E aoss a plane of hage! E 4πσ π K ± ˆ Anothe similaity between eleti and magneti fields. This must be moe than a pue oinidene G. Siolla MT 8.0 etue 10 y 18 9

4 Ampee s law in S n S Ampee s law takes the fom: ids µ 0 whee µ N/A is the magneti pemeability of fee spae enl e aeful not to mi gs and S fomulae! To onvet gs S: multiply by µ 0 /(4π ) Eamples: µ 0 Magneti field eated by a wie: ˆ ϕ ˆ ϕ π Foe between wies: N: fato 1/ missing in F oentz in S F 1 F µ 0 1 π G. Siolla MT 8.0 etue Divegene of ˆ ϕ onside the podued by a wie of uent: alulate its divegene in atesian oodinates: Given y and ˆ ϕ ˆ os y ϕ - ˆ sin ϕ y ˆ - y ˆ + + y + y y ˆ y ˆ y y - i y + y ( + y ) ( + y ) This is a geneal popety of the magneti field: i 0 Simila equation fo E: i E 4π ρ The divegene of E is elated to the density of eleti hages The divegene of must be elated to the density of magneti hages Magneti monopole don t eist (Thee may be magneti monopoles leftove fom the Ealy Univese, but neve obseved epeimentally so fa) G. Siolla MT 8.0 etue

5 Thoughts on What eatly is a magneti field? Why does it have so muh in ommon with eleti field E? Why should thee be a fie d that ats only on moving hages? l Answe: Speial Relativity Relativity: the physis must be the same in all efeene fames A hage at est fo obseve 1 appeas in motion to obseve that moves with a etain veloity w..t. obseve 1: Obseve 1 will measue an eleti field Obseve will measue a magneti field alulating attative o epulsive foe ating on a test hage in the efeene fames will lead to the same onlusion G. Siolla MT 8.0 etue 10 1 Summay and outlook Today: Magneti Field Magneti Foe ating on hages in motion Ampee s aw Net time: Quik ntodution to Speial Relativity Goals: Undestand how and why Magnetism and Eletiity ae elated Finally play with some eally ool physis! G. Siolla MT 8.0 etue 10 11