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Physics 272: Electricity and Magnetism Monday July 2nd Mark Palenik Midterm • Reminder: Thursday at 9:50, 2 hours – We will go over answers afterward • Tomorrow is the last lecture (Wednesday July 4th no class) • No recitation Thursday Midterm topics • • • • Electric field of a point charge Electric field of dipoles (on axis and perpendicular axis) Polarization of materials Electric field and conductors – Field inside a conductor – Field produced by a conductor • • • • Electric field of a uniformly charged sphere (inside and outside) Electric field of a rod (short, infinite) Electric field of a capacitor Electric potential and potential difference – Of a generic electric field – Of a point charge – Go back and forth between E and V • • • • Electrostatic potential energy Energy stored in the electric field Conventional current and electron current Biot-Savart law – Moving charge – Currrent element • Cross products Topics for today • Magnetic field of a current loop • Field of magnetic dipoles – Electric vs. magnetic dipoles – Current loops and dipoles • Atoms as dipoles • Also: A bar magnet is a magnetic dipole, but we’ll talk about that more tomorrow Magnetic field of current distribution • Remember Biot-Savart law: μ0 4𝜋 𝐼×𝑟 𝑑𝑙 2 𝑟 1. Cut up the current distribution into pieces and draw B 2.Write an expression for B due to one piece 3.Add up the contributions of all the pieces 4.Check the result We will also be exploiting some symmetries of the object, so lets think about those Recall again. . . Right hand rule • Thumb in direction of conventional current, fingers wrap in direction of the wire • Electron current and conventional current point in opposite directions. iClicker: electron current • Which direction does the electron current in the wire point? a) b) c) d) e) North South East West The current is zero iClicker • What is the direction of the magnetic field at the x? y a) b) c) d) e) +y -y +z -z 0 z x iClicker: Ring • Remember what we said on the previous slide. If the conventional current is running clockwise in the loop, which way does the magnetic field point at the center? a) b) c) d) e) It is zero +y -y +z -z y I z I I x I x A third right hand rule • Wrap fingers in the direction of the current loop. Thumb points in the direction of B. I I I x I Magnetic Field of a Wire Loop Step 1: Cut up the distribution into pieces Dl = R cos (q + dq ) , Rsin (q + dq ) ,0 - R cosq , Rsin q , 0 → −𝑅 sin 𝜃 , 𝑅 cos(𝜃) 𝑑𝜃 r = 0,0, z - R cosq , R sin q ,0 Make use of symmetry! Need to consider only Bz due to one dl Magnetic field of a Wire Loop • ∆𝑙 × 𝑟 =< −𝑅 sin 𝜃 , 𝑅 cos 𝜃 , 0 > ∆𝜃 ×< 𝑅 cos 𝜃 , 𝑅 sin 𝜃 , 𝑧 > • =< 𝑅 cos 𝜃 𝑧, 𝑅 sin 𝜃 𝑧, −𝑅2 >→ −𝑅2 𝜇0 1 𝜇0 ∆𝑙 𝜇0 ∆𝑙 • ∆𝐵 = ∆𝑙 × 𝑟 2 = = 4𝜋 𝑟 4𝜋 𝑟 3 4𝜋 (𝑧 2 +𝑅 2 )3/2 • 𝑑𝑙 × 𝑟 = 𝑅2 𝑑𝜃 • 𝐵= 2𝜋 𝜇0 𝑅 2 𝑑𝜃𝑧 0 4𝜋 (𝑧 2 +𝑅 2 )3/2 m0 2p R 2 I Bz = 4p ( R 2 + z 2 )3/2 Magnetic Field of a Wire Loop m0 2p R 2 I Bz = 4p ( R 2 + z 2 )3/2 Step 4: Check the results æ T × m ö ( m × A) =T çè ÷ø A (m ) 2 units: 2 3/2 direction: Check several pieces with the right hand rule Note: We’ve not calculated or shown the “rest” of the magnetic field Current carrying ring vs. charged ring • Compare the electric field at the center of a uniformly charged ring to the magnetic field in a current-carrying ring. What do you notice about the field strength at the center of the ring? • What happens as you pass through the center of a current carrying ring vs. charged ring? Magnetic dipole • 𝐵𝑧 = 𝜇0 2𝜋𝑅 2 𝐼 4𝜋 𝑅 2 +𝑧 2 3/2 • z>>R, so 𝑅2 + take the limit as we get very far away 𝑧 2 3/2 → 𝑧3 and 𝜇0 2𝜋𝑅 2 𝐼 4𝜋 𝑅 2 +𝑧 2 3/2 • What does this look like? Like an electric dipole! 1 2p Ez 4 0 z 3 p sq 0 2R 2 I Bz 4 z 3 → 𝜇0 2𝜋𝑅 2 𝐼 4𝜋 𝑍 3 of a loop • From far away, Bz = 𝜇0 2𝜋𝑅 2 𝐼 4𝜋 𝑍 3 • And from an electric dipole, Ez = 1 2𝑃 4𝜋∈0 𝑍 3 • For electric dipoles, we called the dipole moment P. • For magnetic dipoles, the dipole moment is called • What is for a loop? 𝜇0 2𝜋𝑅 2 𝐼 4𝜋 𝑍 3 2 • Bz = • 𝜇 = 𝜋𝑅 𝐼 = 𝜇0 2𝜇 4𝜋 𝑍 3 iClicker question • Which axis do we have to flip on this current ring to reverse the direction of the magnetic field everywhere? y a) X b) Y c) Z z x Reflection of Current Ring y y x x z z B µ v ´ r̂ B ® -B Magnetic vs. Electric dipoles • How do the direction of the electric and magnetic field lines compare in the two dipoles below? • Which way do we have to flip a magnetic dipole? • Which way do we have to flip an electric dipole to reverse the direction of its field at every point? -q +q N Current loops • If we have a bunch of loops sitting on top of each other, we can usually pretend they’re all in exactly the same place. • Field from N loops = N*Field from one loop • Bz = 𝜇0 2𝜋𝑁𝑅 2 𝐼 4𝜋 𝑍 3 I B iClicker of N loops • What is the magnetic moment, , of a current carrying ring with N loops, radius R, and current I? a) b) c) d) 2NR2I NR2I 0NR2I 20NR2I Atoms as dipoles • Remember, current loops are dipoles • Shrink the loop down and you get a point dipole • A point dipole doesn’t exactly have current, since there is no motion (single point) • Electrons can still have angular momentum. • Electrons have “spin” angular momentum and behave as point dipoles. • Can only know one component which can take on 2 values Atoms as dipoles • Electrons can also have orbital angular momentum around the nucleus • You can think of it like a classical orbit (current) • This also produces a magnetic field, like a regular current loop.