16.1 Permanent magnets

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Roderick Bryan
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Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops

to as its noth pole; the othe end is its south pole The ule fo whethe two magnets attact o epel each othe: opposites attact; likes epel Beaking a magnet in half esults in the appeaance of two new

1 Unit 16 Magnetism 161 Pemanent magnets 16 The magnetic foce on moving chage 163 The motion of chaged paticles in a magnetic field 164 The magnetic foce exeted on a cuent-caying wie 165 Cuent loops and magnetic toque 166 Biot and Savat s law 161 Pemanent magnets A ba magnet can attact anothe magnet o epel it, depending on which ends of the magnets ae bought togethe One end of a magnet is efeed to as its noth pole; the othe end is its south pole The ule fo whethe two magnets attact o epel each othe: opposites attact; likes epel Beaking a magnet in half esults in the appeaance of two new poles on eithe side of the beak This behavio is fundamentally diffeent fom that in electicity, whee the two types of chage can exist sepaately We saw a visual indication of the electic field E of a point chage using gass seed suspended in oil Similaly, the magnetic field B can be visualized using small ion filings spinkled onto a smooth suface The filings ae bunched togethe nea the poles of the magnets This is whee the magnetic field is most intense The diection of the magnetic field, B, at a given location is the diection in which the noth pole of a compass points when placed at that location In geneal, magnetic field lines exit fom the noth pole of a magnet and ente at the south pole 1

16 The magnetic foce on moving chage The magnetic foce depends on seveal factos: The chage of the

the velocity vecto and the magnetic field vecto, θ The mathematical elation of them in vecto fom

when θ = 9 o The foce vanishes when θ = o Now we define the magnetic field B as B = F qv sinθ The

especially when compaed with the magnetic field at the suface of the Eath, which is oughly 5 1 5 T

2 16 The magnetic foce on moving chage The magnetic foce depends on seveal factos: The chage of the paticle, q; The speed of the paticle, v; The magnitude of the magnetic field, B; The angle between the velocity vecto and the magnetic field vecto, θ The mathematical elation of them in vecto fom is F = q( v B) One can ewite it as a scala expession, eg F = qvb sinθ The maximum foce is obtained when θ = 9 o The foce vanishes when θ = o Now we define the magnetic field B as B = F qv sinθ The SI unit is 1 tesla = 1 T = 1 N/(A m) The tesla is a faily lage unit of magnetic stength, especially when compaed with the magnetic field at the suface of the Eath, which is oughly T Thus, anothe commonly used 4 unit of magnetism is the gauss (G), defined as follows: 1G = 1 T In tems of the

3 gauss, the Eath s magnetic field on the suface of the Eath is appoximately 5 G A ba magnet has a magnetic field of oughly 1 G Remak: Magnetic field lines neve coss one anothe As the diection in which a compass points at any given location is the diection of the magnetic field at that point Since a compass can point in one diection, thee must be only one diection fo the field B If field lines wee to coss, howeve, thee would be two diections fo B at the cossing point, and this is not allowed Example Paticle 1, with a chage q 1 = 36 C and a speed v 1 = 86 m/s tavels at ight angles to a unifom magnetic field The magnetic foce it expeiences is N Paticle, with a chage q = 53 C and a speed v = m s moves at an angle of 55 o / elative to the same magnetic field Find (a) the stength of the magnetic field and (b) the magnitude of the magnetic foce exeted on paticle Answe Apply the fomula: Apply the fomula: F = qvb sinθ with θ = 9 o One obtains 3 F 45 1 N B = = = 137T 6 qvsin θ (36 1 C)(86 m/ s)sin 9 o F = qvb sinθ with θ = 55 o One obtains 6 3 o F (53 1 )(13 1 m/ s)(137 T) sin N = = The diection of the magnetic foce is given by the magnetic foce ight-hand ule (RHR), which states as follows To find the diection of magnetic foce on a positive chage, stat by pointing the finges of you ight hand in the diection of the velocity, v Now cul you finges fowad the diection of B You thumb points in the diection of F 3

If the chage is negative, the foce points opposite to the diection of you thumb Note that the magnetic foce F points in a diection that is pependicula to both B and the chage velocity v As an

4 If the chage is negative, the foce points opposite to the diection of you thumb Note that the magnetic foce F points in a diection that is pependicula to both B and the chage velocity v As an example, the diection of a magnetic foce F can be indicated by the magnetic foce RHR extending ou finges to the ight (v) and then culing them into the page (B) we see that the magnetic foce exeted on this paticle is upwad, as indicated If the chage is negative, the diection of F is evesed Example Thee paticles tavel though a egion of space whee the magnetic field is out of the page as shown in figue Fo each of the thee paticles, state whethe the paticle s chage is positive, negative, o zeo Answe Use the RHR to obtain the following Paticle 1: negative; paticle : zeo and paticle 3: positive 4

163 The motion of chaged paticles in a magnetic field Figues (a) and (b) shows the motion of a positive chaged paticle moving unde electic field and magnetic field espectively Remaks: 1 As the

5 163 The motion of chaged paticles in a magnetic field Figues (a) and (b) shows the motion of a positive chaged paticle moving unde electic field and magnetic field espectively Remaks: 1 As the diection of velocity and the diection of magnetic foce ae pependicula to each othe, the wok done by the magnetic foce is always zeo A chaged paticle with a velocity v that is pependicula to the magnetic field moves in a cicula path The magnetic foce acts as the centipetal foce, eg = and F cp F cp = qvb Hence = qvb gives the adius of cicula path, = On the othe hand, F cp = = mω = qvb, qb we can wite m ω = qb, as v = ω Plugging in the elation ω π π =, we have m = qb T T πm T = qb, and the peiod T is given by 3 Helical motion is a combination of linea motion and cicula motion 5

6 Example In a device called a velocity selecto, chaged paticles move though a egion of space with both an electic and a magnetic field If the speed of the paticle has a paticula value, the net foce acting on it is zeo Assume that a positively chaged paticle moves in the positive x diection, and the electic field is in the positive y diection Should the magnetic field be in (a) the positive z diection, (b) the negative y diection, o (c) the negative z diection in ode to give zeo net foce? Answe The foce exeted by the electic field is in the positive y diection; hence, the magnetic foce must be in the negative y diection if it is to cancel the electic foce If we simply ty the thee possible diections fo B one at a time, applying the magnetic foce RHR in each case, we find that only a magnetic field along the positive z axis gives ise to a foce in the negative y diection, as desied The answe is (a) Example Two isotopes of uanium, 35 5 U ( 39 1 kg ) and 38 5 U ( kg ), ae sent 5 into a mass spectomete with a speed of 15 1 m/ s Given that each isotope is singly ionized, and that the stength of the magnetic field is 75 T, what is the distance d between the two isotopes afte they complete half a cicula obit? Answe The isotopes ae singly ionized, which means that a single electon has been emoved 19 fom each atom And the isotopes ae now having chage of e = 16 1 C 6

7 Fom the elation qb = qvb, we obtain v =, and thus = m qb 5 5 (39 1 kg)(15 1 m / s) = = = 341cm qb (16 1 C)(75 T) 5 5 (395 1 kg)(15 1 m / s) = = = 346cm qb (16 1 C)(75 T) The sepaation between the isotopes: d = = (346cm 341 cm) = 1cm 164 The magnetic foce exeted on a cuent-caying wie As a chaged paticle expeiences a foce when it moves acoss magnetic field lines The same thing happens when a wie caies cuent on it Conside a staight segment of length L of a wie with a cuent I flowing fom left to ight, pesents in a magnetic field B, as shown in figue If the conducting chages move though the wie with an aveage speed v, the time equied fo them to move fom one end of the wie segment to the othe is t = L / v The amount of chage that flows though the wie in this time is q = I t = IL / v Theefoe, the foce exeted on the wie is I L F = qvbsinθ = vbsinθ v Hence, we have F = ILB sinθ Maximum foce occus when the cuent is pependicula to the magnetic field (θ = 9 o ) and is zeo if the cuent is in the same diection as B (θ = o ) The diection of the magnetic foce is given by the RHR, whee the diection of chage velocity v is now the diection of cuent I Example When the switch is closed in the cicuit, the wie between the poles of the hoseshoe magnet deflects downwad Is the left end of the magnet (a) a noth magnet pole of (b) a south magnetic pole? 7

Answe Once the switch is closed, the cuent in the wie is into the page, as shown in the ight figue Applying the magnetic foce RHR, we see that the magnetic

the left end of the magnet must be a noth magnetic pole The answe is (a) 165 Cuent loops and magnetic toque A toque is expeienced by a cuent loop as shown in

clockwise w w τ = ( IhB ) + ( IhB) = IB( hw) = IAB, Plug in the aea of the ectangula loop, A = hw We haveτ = IAB If the plane of loop makes an angle with the

8 Answe Once the switch is closed, the cuent in the wie is into the page, as shown in the ight figue Applying the magnetic foce RHR, we see that the magnetic field must point fom left to ight in ode fo the foce to be downwad Since magnetic field lines leave fom noth poles and ente at south poles, it follows that the left end of the magnet must be a noth magnetic pole The answe is (a) 165 Cuent loops and magnetic toque A toque is expeienced by a cuent loop as shown in figue If we imagine an axis of otation though the cente of the loop, at the point O, it is clea that the foces exet a toque that tends to otate the loop clockwise w w τ = ( IhB ) + ( IhB) = IB( hw) = IAB, Plug in the aea of the ectangula loop, A = hw We haveτ = IAB If the plane of loop makes an angle with the magnetic field, we haveτ = IABsinθ Fo the case that has n tuns in a geneal loops, we haveτ = niabsinθ Applications of the magnetic toque ae electic motos and galvanomete 8

166 Biot and Savat s law Chage E Electic field Chage Cuent B field Cuent Recall that in the Coulomb s law, we have, say, a point chage q 1 gives an electic field at P and E = ε q distibution, the

9 166 Biot and Savat s law Chage E Electic field Chage Cuent B field Cuent Recall that in the Coulomb s law, we have, say, a point chage q 1 gives an electic field at P and E = ε q distibution, the electic field is at P due to dq is ˆ Fo a chage 1 de = ε dq ˆ dq de Now, fo a magnetic field, we have the law of Biot and Savat I ds sinθ db =, 7 7 whee = 1 T m/ A= 1 H / m ( Heny = H = T m A ) and is called the / pemeability constant In vecto fom, we I ds ˆ have db = Compaing with the Coulomb s 1 law, we ealize that both ae laws I ds θ I ˆ Magnetic field P Example Find the magnetic field at the cente of a coil which caies a steady cuent I Answe The magnetic field though the cente of coil is the supeposition of the magnetic field due to cuent segment I ds sinθ Hence, B = As is a constant and θ = 9 o, we can wite I I ( ) I B = ds π = = NI If the coil has N tuns, the magnetic field is given by B = 9

Remak: If the solenoid has N tuns, length l and caies a cuent I, the magnetic field B at a point O on the axis nea the cente of the solenoid is found to be B NI = = ni, l whee n is the numbe of tuns

10 Remak: If the solenoid has N tuns, length l and caies a cuent I, the magnetic field B at a point O on the axis nea the cente of the solenoid is found to be B NI = = ni, l whee n is the numbe of tuns pe unit length of coil If the coil is infinite long o long enough that its length is ten times the diamete, the magnetic field at the end of coil is half that at the cente of coil, eg ni B = Example (Challenging) Find the magnetic field B at a point P, a distance z fom a long cuent wie, as shown in the figue Answe Fom Biot-Savat s law, we have I dl ˆ B =, whee dl ˆ = dl sinφ = dl cosθ As l = ztanθ, we have dl = ( z / cos θ) dθ π I dl cosθ We can wite B = π Now, z/ = cosθ 1 cos θ = gives z B = I π z cos θ π cos θ z π π π cosθ dθ sinθ π cosθ dθ I I I = = = z z πz I P z O θ l φ dl I Hence, the magnetic field at P due to a long wie of cuent is given by B = π z 1

11 Remak 1: The magnitude of the magnetic field 1 m fom a long, staight wie caying a cuent 7 I ( 1 T m/ A)(1 A) 7 of 1 A is B = = = 1 T πz π(1 m) This is a weak field, less than one hundedth the stength of the Eath s magnetic field Remak The magnetic field shown in the figue is due to the hoizontal, cuent-caying wie The cuent in the wie flows to the left The following states the use of magnetic field ight-hand ule which points the diection of cuent: If you point the thumb of you ight hand along the wie to the left you finges cul into the page above the wie and out of the page below the wie Thus, the cuent flows to the left 11

MAGNETIC FIELD INTRODUCTION

MAGNETIC FIELD INTRODUCTION It was found when a magnet suspended fom its cente, it tends to line itself up in a noth-south diection (the compass needle). The noth end is called the Noth Pole (N-pole),