The Magnetic Field. Part 1. Magnetism. Chapter 20. What creates magnetic fields? Unlike electrostatics: Magnetic monopoles have never been detected.

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Bertha Dorsey
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1 The Magnetic Field Magnetism Chapte 0 Pat 1 What ceates magnetic fields Unlike electostatics: Magnetic monopoles have neve been detected. Thee is no magnetic chage! Any pemanent magnet has two poles. What if to cut a pemanent magnet in half Inteaction between magnetic poles Opposite magnetic poles attact each othe, and like poles epel each othe Magnetic Field Lines Assume that thee is a magnetic field B in some aea of space We can epesent magnetic fields with field lines, as we did fo electic fields (1) the diection of the tangent to a magnetic field line at any point gives the diection of B at that point () the spacing of the lines epesents the magnitude of B 1

2 Magnetic Field Lines (cont.) The Eath s Magnetic Field The field lines ente one end of a magnet and exit the othe end. The end of a magnet fom which the field lines emege is called the noth pole of the magnet the othe end, whee field lines ente the magnet, is called the south pole The spinning ion coe of the eath poduces a magnetic field. The magnetic noth pole coesponds to the geogaphic south pole. South Pat Magnetic Foce on a Paticle Magnetic foce on a chaged paticle F q electic chage B = q v B v paticle velocity B magnetic field F B = q vb sinφ the magnitude Right Hand Rule / Teddy Bea Rule / Alligato Rule Right Hand Rule: Positive and negative paticle = q v B F B Magnetic field (definition) F B = q vb sinφ FB B = q v q electic chage v paticle velocity B magnetic field SI unit: Tesla 1 T = newton/(c*m/s) = 1 N/(A*m) 1 tesla = 10 4 gauss (G) The foce acting on a chaged paticle moving though a magnetic field is always pependicula to the velocity and the field

3 Notation To depict a vecto oiented pependicula to the page we use cosses and dots. A coss indicates a vecto going into the page (think of the tail feathes of an aow disappeaing into the page). A dot indicates a vecto coming out of the page (think of the tip of an aow coming at you out of the page). B out of the page B into the page Diection of Magnetic Foces (cont.) The diection of the magnetic foce on a moving chage is pependicula to the plane fomed by B and v. F Β v To detemine the diection, you must apply the Right Hand Rule (RHR). A magnetic field exets a foce on a chaged paticle: A) always B) neve C) if the paticle is moving acoss the field lines D) if the paticle is moving along the field lines E) if the paticle is at est F B = q vb sinφ Example A chage of 3 mc is moving in the negative x diection at 4 m/s. A magnetic field of 30 T is pointing in the positive y diection. What is the magnitude and diection of the foce on the chage How does you answe change if the chage is 3 mc 6 o F = qvbsinθ = sin 90 3 F = N Use the ight-hand ule! Diection is zˆ, i.e. into the pape If the chage is negative, Diection is zˆ. poblem Motion of Chages in B Fields Pat 3 The Motion in a Magnetic Field If a chaged paticle is moving in a diection pependicula to a unifom magnetic field, then its tajectoy will be a cicle because the foce F=qvB is always pependicula to the velocity, and theefoe centipetal. F c = ma= Recall that so F = qvb= The adius of the cicula tajectoy = qb π π m T = = The peiod (the time fo one full evolution) v qb 3

4 F qvb = = An electon and a poton ae both initially moving with the same speed and in the same diection at 90 0 to the same unifom magnetic field. They expeience magnetic foces, which ae initially: A) identical B) equal in magnitude but opposite in diection C) in the same diection and diffeing in magnitude by a facto of 1840 D) in opposite diections and diffeing in magnitude by a facto of 1840 E) equal in magnitude but pependicula to each othe An electon and a poton each tavel with equal speeds aound cicula obits in the same unifom magnetic field, as shown in the diagam (not to scale). The field is into the page on the diagam. (a) Whee is the electon (b) What is the diection = qb A unifom magnetic field is diected into the page. A chaged paticle, moving in the plane of the page, follows a clockwise spial of deceasing adius as shown. A easonable explanation is: A) the chage is positive and slowing down B) the chage is negative and slowing down C) the chage is positive and speeding up D) the chage is negative and speeding up E) none of the above Cosmic ays (atomic nuclei stipped bae of thei electons) would continuously bombad Eath s suface if most of them wee not deflected by Eath s magnetic field. Given that Eath is, to an excellent appoximation, a magnetic dipole (a ba magnet), the intensity of cosmic ays bombading its suface is geatest at the 1. poles.. mid-latitudes. 3. equato. = qb Isotope Sepaation = qb Cossed E and B fields 4

5 Foce on a Cuent Caying Wie Pat 4 Magnetic Foce on a Cuent Cuent in a wie is a collection of moving chages; theefoe, a cuent caying wie in a magnetic field also expeiences a foce. If a wie of length L, caying a cuent I, makes an angle θ with a magnetic field B, then the magnitude of the foce on the wie is: F = qvbsinϑ = ( It) vbsinϑ = L = ( I ) vbsinϑ = ILBsinθ v F = ILBsinθ Β θ Ι F Foce on a Cuent Caying Wie F = ILBsinθ Foce on a Cuent Caying Wie F = ILBsinθ The diagam shows a staight wie caying a flow of electons into the page. The wie is between the poles of a pemanent magnet. The diection of the magnetic foce exeted on the wie is: A) B) C) D) E) into the page Magnetic Levitation (Maglev, etc.) ILB= mg Magnetic Toque on cuent loop In a unifom magnetic field, the net foce on a cuent loop (independent of geomety) is 0. Howeve, thee can be a toque τ = F sinθ = distance fom axis of otation to loop segment F = magnetic foce on segment θ = angle between vecto and vecto F. τ = w w IhB + IhB = IB( hw) τ = IBA Only the vetical segments of the loop expeience a foce. The toque will otate the loop so that the plane of the loop is pependicula to the magnetic field. 5

6 Magnetic Toque on cuent loop (cont.) Toque exeted on a ectangula loop of aea A τ = IBAsinθ If the loop has N tuns, then τ = NIBAsinθ A squae loop of wie lies in the plane of the page and caies a cuent I as shown. Thee is a unifom magnetic field paallel to the side MK as indicated. The loop will tend to otate: A) about PQ with KL coming out of the page B) about PQ with KL going into the page C) about RS with MK coming out of the page D) about RS with MK going into the page E) about an axis pependicula to the page Pat 5 Magnetic Fields Due to Cuents Expeimental obsevation in 180 Hans Oested: Electic cuents can ceate magnetic fields The magnitude of the field poduced at point P A geneal equation fo Physics 3 μ0 idssinϑ db = 4π 6 μ = T m / A 0 A Long Staight Wie Poof: integation o Ampee s law μ0i B = πr μ = T m / A 6

7 Conceptual question The equation above is tue fo an infinitely long, staight conducto caying a cuent. Of couse, thee is no such thing as an infinitely long anything. How would you decide whethe a paticula wie is long enough to be consideed infinite μ0i B = Poblem πr poblem Conside the long, staight, cuent-caying wies shown in the figue. One wie caies a cuent of 6. A in the negative y diection; the othe caies a cuent of 4.5 A in the positive x diection. Calculate the magnitude and diection of the net magnetic field at points A and B. Foce between two paallel cuents Each of two paallel wies with cuents I 1 and I, expeiences a magnetic foce given by Magnitude of B a μ0ia Ba = πd Foce on a length L of wie b F = i LB ba b a F ba μ0liai = πd b F = μ 0 I 1 I π d L = length of wie d = distance between the two wies If the cuents ae paallel, the foce is attactive. If the cuents ae anti-paallel the foce is epulsive. L Two long paallel staight wies cay equal cuents in opposite diections. At a point midway between the wies, the magnetic field they poduce is: A) zeo B) non-zeo and along a line connecting the wies C) non-zeo and paallel to the wies D) non-zeo and pependicula to the plane of the two wies E) none of the above On a compute chip, two conducting stips cay chage fom P to Q and fom R to S. If the cuent diection is evesed in both wies, the net magnetic foce of stip 1 on stip 1. emains the same.. eveses. 3. changes in magnitude, but not in diection. 4. changes to some othe diection. 5. othe 7

8 Conceptual question Steams of chaged paticles emitted fom the sun duing unusual sunspot activity ceate a distubance in the eath s magnetic field (called a magnetic stom). How can they cause such a distubance B Fields of Cuent Distibutions By winding wies in vaious geometies, we can poduce diffeent magnetic fields. Fo example, a cuent loop (pependicula to plane, adius R, cuent emeging fom plane at top of loop): Magnitude of magnetic field at the cente of loop: Ι Nμ I B = 0 R N = # of loops of wie (i.e. # tuns) Diection of magnetic field fom the RHR. Solenoids If we stack seveal cuent loops togethe we end up with a solenoid. In the limit of a vey long solenoid, the magnetic field inside is vey unifom: B = μ ni 0 n = numbe of windings pe unit length, I = cuent in windings Magnetic Mateials On atomic level moving electons (micoscopic cuent loops) ceate magnetic fields. In many mateials, these cuents ae andomly oiented (net magnetic field is zeo). In some mateials, the pesence of an extenal magnetic field can cause the loops to become oiented Paamagnetism oientation with an extenal field Diamagnetism oientation against an extenal field Feomagnetism line up loops (magnetic domains) without an extenal field 8

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent