B-1 Magnetism: a new force! So far, we've learned about two forces: gravity and the electric field force. EFEq=GG , EEFq=GG ← Definition of E-field • E-fields are created by charges: 2kQ|E|r=G • E-field exerts a force on other charges: EEFq=GG. The ated by mass: gr vitational force is similar: E 2GM|g|r=G• Gravitational fields are crea. • The gravitational field exerts a force on other masses. ggravFm=GG. here is a different kind of field, called a magnetic field or B-field. The repends on the velocity of the charge in a T• B-fields are created by moving charges (currents). • B-fields exert forces on moving charges. fo ce FB from a B-field on a moving charge dpeculiar way: a charge q, moving with velocity vG in a magnetic field BG, feels a force of magnitude BBFFqvBsin qvB⊥== θ =G If q B θ B θ v v v⊥ = v sinθ vB⊥GG, then sinθ = 1 ⇒ FB = |q| v B If then sinθ = 0 ⇒ FvBGG&,B = 0 If v = 0, then FB = 0 (unlike gravity or E-field force) The direction of the force BFGis perpendicular to the pla vG and . Direction of BGBFG ne formed by is determined by the "Right-hand rule". Use your right hand. Point your fingers in the direction Last update: 3/1/2006 Dubson Phys2020 Notes, ©University of ColoradoB-2 of vG, curl your fingers toward BG. Thumb then points in the direction of BFG if the charge q ipositive. (Orient hand so that fingers curl thru angle < 180). If q is (-), BFs oG is other way v B FB q B v F (out) q B v F (in) q (–) B F (in) [F] NUnits of B: [B] = [B] 1 t a (T)[q][Cs=⋅ of B: 1 gauss = 10-4 T , 1 T = 104 gauss -5kitchen magnet : 50 – 500 gauss = 0.005 – 0.05 T ough to yank tools of your hand.) Currents make B-fields ake a B-field with a currenteslmv]==Older, non-SI, unit• Earth's magnetic field ≈ 0.5 gauss = 5 × 10 T • • iron core electromagnet: 2 T (max) (Strong en out• superconducting magnet: 20 T (max) You m I. (Charges make E-fields, currents make B-fields). ire, carrying current I, has magnitude The B-field produced by a long straight w0IB2rµ=π r = distance from wire, µo = constant = 4π×10-7 (SI units) v (This formula can be derived from a fundamental law called Ampere's Law, which relates curren ts and B-fields .) Last update: 3/1/2006 Dubson Phys2020 Notes, ©University of ColoradoB-3 The direction of the B-field is perpendicular to the direction of the current. The B-field lines form circles around the current, according to "Right-hand Rule II": Point thumb of RH along current I, ngers curl along direction of B-field. t complicated B-field: -field lines are fundamentally different from E-field But B-field lines always form losed loops with no beginning or end. with a fi A loop of wire carrying a curren I forms a rather Blines in this way: E-field lines begin and end on charges (or go to ∞ ). c It is possible to make a uniform, constant B-field solenoid = cylindrical coil of wire: I I B End View: I B uniform inside Side View I (out) B r B B B I B I B Last update: 3/1/2006 Dubson Phys2020 Notes, ©University of ColoradoB-4 Motion of a charged particle in magnetic field Consider a charge q moving in a uniform magnetic field B. Since the force ys perpendicularFB is alwa to the velocity v, the force FB does no work : FFv W 0=B⊥⇒BGG. The magnetic force cannot change the KE or PE of the particle (since Work done = ∆K The B-field changes the direction of t, but does not change the speed. the velocity v is perpendicular to the field B, the magnetic force bends the path of the particle We can relate the radius R of the circular path to the magnitude of the field B and the speed v with Newton's Second Law: E + ∆PE). he velocity vIfin a circle. 2netvFma qvBmR= ⇒ =GG ( recall that for circular motion 2vaR=G) Solving for R, we get mvRqB= . Notice that the radius is proportional to the mass of the particle. In a mass-spectrometer, the mass of an unknown particle is determined from measurement of the radius (charge, speed and B-field are all known). The Velocity Selector The velocity selector is a device which measures the speed v of an ion. (ion = charged atom with one or more electrons missing) . A magnet produces a uniform B-field and a capacitor produces a uniform E-field, with E ⊥ B . v FB uniform B (out) v FB v FB RB(out)q q v FB Last update: 3/1/2006 Dubson Phys2020 Notes, ©University of ColoradoB-5 The B and E fields are adjusted until the particle goes straight through. If the path is straight, then FB = FE ⇒ qvB = qE ⇒ v = E / B . Magnetic force on a current-carrying wire A B-field exerts a force on a moving charge. A current-carrying wire is full of moving charges, so a B-field exerts a force on the current-carrying wire. Recall thBsin⊥== θ . We will show below that this leads to where I is the current, L is the length of the wire, and θ is the angle between the direction of B and the direction of th current. roof of Fon wire = I L B: Consider a section of wire, length L, carrying N moving charges. at on qFqvBqvon wireFILBsin= θ ePAssume B ⊥ wire. speed of moving charges = Lvt= current NqIt= NNon wire on qLItFNFNqvB LBILBt= ⋅ == = (done) NqB (uniform) L I F (in) F = I L B ( B⊥I direction) Forces on chFE = q E arge: FB = q v B E B(in) v mass m charge q uniform E down, B (in) L v N moving charges B Last update: 3/1/2006 Dubson Phys2020 Notes, ©University of ColoradoB-6 Current-carrying wires exert magnetic forces on each other. Wire1 creates a B-field ( 0IB2rπµ= ). Wire2 feels a force due to the B-field from wire A (LB). Parallel currents attract, Anti-parallel currents repel. (Can you see why? Use Right hand rule.) e electron is a ge, forming a tiny current loop –– an "atomic current". on wireFI= Permanent Magnets Currents make B-fields. So where's the current in a permanent magnet (like a compass needle)? An atom consists of an electron orbiting the nucleus. Thmoving charIn most metals, the atomic currents of different atoms have random orientations, so there is no net current, no B-field. In ferromagnetic materials (Fe, Ni, Cr, some alloys containing these), the atomline up to produce a large net current. ic currents can all In a magnetized iron bar, all the atomic currents are a sulting in a large net current around the rim of the bar. The current iron bar then acts like a solenoid, producing a uniform B-field inside:
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