Physics 11bLecture #12Sources of the Magnetic FieldS&J Chapter 30What We Did Last Time Lorentz force on current Simplifies to if the B field is uniform Torque on a current loop in a B field Current loop looks like a magnetic dipole Hall effect Magnetic fields are created by electric current Biot-Savart Law B field by an infinite straight-line currentbBaId=×∫FsBBI=×FLBI=×τ ABI=µAHHRIBVt∆=1HRnq≡02ˆ4Iddrµπ×=srB02IBaµπ=Today’s Goals Continue with Biot-Savart law Calculate B field created by a straight-line current Force between two currents Ampère’s Law Integral form of Biot-Savart law Useful in solving magnetic field problems Examples: thick wire, infinite current sheet, solenoid Magnetic flux and Gauss’s law of magnetism Magnetism in matterBiot-Savart Law B field due to current in a short piece ofwire is given by the Biot-Savart Law dB is perpendicular to both ds and r Size of dB depends on the angle between ds and r02ˆ4Iddrµπ×=srBdsdBrIdsIθ0dB=2Idsr2sinIdsrθStraight-Line Current Current I flows in an infinitely-long straight wire B field at distance a from the wire? We move ds from x = −∞ to +∞ r and θchanges as we go We can integrate with x, r, or θÆ which one is the easiest?dsIθdBaxyz0022ˆsinˆ44Id Idxdrrµµθππ×==srBzr222rax=+sinarθ= tanaxθ=−Straight-Line Current Easiest to integrate if switch from dx to dθ Also020sin4sin4IdxdBrIdaµθπµθθπ==sinarθ=tanaxθ=−[]00000sin cos442IIIBdaaaππµµµθθ θπππ==−=∫2sinaddxθθ=B field goes around the wire and decrease as 1/axdsIθdBaxyzrB Field Around Current B field circles around (infinitely-long) current Direction of B follows right-hand rule Magnitude B decreases as cf. E field created by infinitely-longlinear charge distribution was Similar, but E points outward, B rotates aroundBI02IBrµπ=r02Erλπε=charge density (C/m)Force Between Currents Run two straight wires in parallel I1creates rotating B field I2feels the force Parallel currents attract each other If I1and I2flow in opposite directions, they repel each other For and I2I1a012IBaµπ=01222IIFIaµπ=×=LBAAF121AII== 1ma==A270(1A) (1m)210N2(1m)Fµπ−==×This is in fact howAmpere is definedAmpère’s Law The denominator of looks like the circumference of a circle If we draw a circle around thecurrent and line-integrate This is generalized to Ampère’s Law02IBrµπ=r002IddsIrµµπ⋅= =∫∫BsvvdsFor any closed path, the line integral of the magnetic field iswhere I is the total current passing through any surface bounded by the closed path0dIµ⋅=∫BsvAmpère’s Law Ampère’s Law follows Biot-Savart Law Same way as Gauss’s Law follows Coulomb’s Law It’s useful when a symmetry of the problem helps us to predict at least the direction of the B field Example: long wire with finite thickness Current I flows uniformly through a cylindrical wire What’s the B field outside and inside of the wire?RIThick Wire Inside the wire Circle with radius encirclescurrent  Ampère’s Law Outside the wire Circle with radius encircles I Ampère’s LawRr1r21rR<221122rIrIRRππ=211022IrdrBRπµ⋅= =∫Bsv0122IBrRµπ=2rR>202drBIπµ⋅= =∫Bsv022IBrµπ=Same result as athin wireThick Wire Combining, we findRr1r202022IrrRRBIrRrµπµπ⎧≤⎪⎪=⎨⎪>⎪⎩()BrrR02IRµπThis part is sameas thin wires1Br∝Br∝Infinite Current Sheet Infinite sheet carries current density JsA/m B fields are parallel to the sheetopposite directions aboveand below Draw a rectangle andapply Ampère Uniform B field is createdBBA02sdB Iµ⋅= =∫Bs AAv02sIBµ=Two Infinite Current Sheets What if we had two current sheets? Each sheet makes uniform Bfields above and below Add up Æ Uniform B field onlybetween the sheets This looks similar to a capacitor Charged sheet makes E fieldsabove and below Two oppositely-charged sheetsmake uniform E field between Infinite current sheet is lesspractical to make…02sIBµ=0 sBIµ=0B=0B=Solenoid Bend the infinite current sheet into a tube Current flows around the tube Uniform B field inside? We can make this by windinga long wire around a cylinder Suppose the winding is dense Also suppose the solenoid is very long Use Ampère’s Law to find out BNn =ATurns of windingSolenoid lengthi.e. largeSolenoid Field B field is parallel to the solenoid’s axis Also, if you go far away, B Æ 0 Apply Ampère to the rectangular path Line integral doesn’t changeas long as the top and bottomsides of the rectangle remainoutside and insideÆ B field is uniformB = 0B0dB Inµ⋅= =∫Bs AAvA0BnIµ=Magnetic Flux We can define magnetic flux just like the electric flux Integral is taken over a given surface area “How many magnetic field lines go through this area?” Units are T·m2 = N·m/A for magnetic flux V·m for electric flux Main use of the electric flux was Gauss’s Law What happens to it with magnetic field? BdΦ= ⋅∫BA EdΦ=⋅∫EAGauss’s Law With electric flux, we had But there is no magnetic “charge” Gauss’s Law in magnetism: i.e., magnetic flux through any closed surface is zero Not as useful (in solving problems) as the electric version But we’ve got Ampère’s Law instead Magnetic flux is a critical ingredient for Faraday’s Law We’ll get to that next weekin0EqdεΦ= ⋅ =∫EAv0BdΦ=⋅=∫BAvMagnetism in Matter Different materials react differently to magnetic field Some (e.g. iron) stick and others don’t Read textbook 30.8 Things get a bit encylopedic Three types of magnetism Ferromagnetism – strongly attracted to magnets Iron (ferrrum), cobalt, nickel, etc. Paramagnetism – weakly attracted to magnets Ferromagnetic metals turn paramagnetic at high temperature Diamagnetism – weakly repelled by magnets Majority of materials Special case: superconductors are strongly diamagneticSummary Applied Biot-Savart law to linear current B field rotates around the current Parallel wires attract each other by Definition of Ampere Ampère’s Law Applies to any boundary that encircle current I Examples: thick wire, infinite current sheet Solenoid field Gauss’s Law Ferro-/para-/diamagnetism in