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

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1 Foces & Magnetic Dipoles x θ F θ F. = AI τ = U =

2 Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet

3 Faaday s moto Wie with cuent otates aound a Pemanent magnet I df = I dl

4 Toque on a ectangula cuent loop The loop will feel a toque which tends to otate the loop, back into plane Conside a wie loop dimensions a x b whose plane is an angle φ elative to a constant field. Thee will be a net toque whose magnitude on this loop is given by, τ = Iab sinφ = I ( aea) sinφ

5 I Magnetic Moment,, of a ectangula cuent loop Definition ; = cuent aea= I A is vecto quantity, whose diection is nomal to loop plane, use ight hand ule to define diection. We can define vecto toque moe conveniently in tems of vecto magnetic moment cossed by field τ = Anothe name fo a cuent loop is magnetic dipole

6 Deivation of Foces on wies of length b ae collinea and cancel. Foces on wies of length a cancel but ae not collinea. Thee will be a toque. F F = I l = I l = Ia τ = Toque on a ectangula cuent loop b τ = 2 F = 2( sinφ)( Ia) 2 τ = I( ab)sinφ τ = sinφ = τ = ϕ = b/2 ϕ F

7 Peflight 13: F v I L A squae loop of wie is caying cuent in the counteclockwise diection. Thee is a hoizontal unifom magnetic field pointing to the ight. 2) What is the foce on section a-b of the loop? a) zeo b) out of the page c) into the page 3) What is the foce on section b-c of the loop? a) zeo b) out of the page c) into the page 4) What is the net foce on the loop? = a) zeo b) out of the page c) into the page ab: F ab = 0 = F cd since the wie is paallel to. bc: F bc = IL RHR: I is up, is to the ight, so F points into the sceen. y symmety: F da = F bc F F F F F net = ab + bc + cd + da = 0

8 Magnetic Moment,, of abitay loop Definition ; = cuent aea= I A We can moe geneally define magnetic moments only by aea and do not need to know the actual dimensions. This is geneal I I I Fo any shape, = cuent aea= I A and τ τ = AI sinθ = Note: if loop consists of N tuns, = NAI

9 Toque on loop in diffeent angula positions τ I Net toque is zeo when is paallel to I Net toque is maximum when is pependicula to

10 a Magnet Analogy You can think of a magnetic dipole moment as a ba magnet: = N In a magnetic field they both expeience a toque tying to line them up with the field As you incease I of the loop stonge ba magnet We will see that such a cuent loop does poduce magnetic fields, simila to a ba magnet.

11 Application; galvanomete uses toques on coiled loops in magnetic field Toque is poduced about the needle axis and this counte acts the estoing sping and enables the needle to otate Cuent inceased = I Aea inceases Toque fom inceases Angle of needle inceases Cuent deceased deceases Toque fom deceases Angle of needle deceases This is how almost all dial metes wok voltmetes, ammetes, speedometes, RPMs, etc.

12 Example: Loop in a -Field A cicula loop has adius R = 5 cm and caies cuent I = 2 A in the counteclockwise diection. A magnetic field =0.5 T exists in the negative z-diection. The loop is at an angle θ = 30 to the xy-plane. y z x x x x x x x x x x x x x x I x x x x x x x x x x x x x x z y X θ x X x What is the magnetic moment of the loop? = π 2 I =.0157 Am 2 The diection of is pependicula to the plane of the loop as in the figue. Find the x and z components of : x = sin 30 =.0079 Am 2 z = cos 30 =.0136 Am 2

13 Electic Dipole Analogy +q F τ = F τ = F F = p qe F. -q E x θ θ F= IL p = 2 qa = NAI F F. (pe tun) τ = p E τ =

14 Peflight 13: A squae loop of wie is caying cuent in the counteclockwise diection. Thee is a hoizontal unifom magnetic field pointing to the ight. 6) What is the net toque on the loop? a) zeo b) up c) down d) out of the page e) into the page τ = points out of the page (cul you finges in the diection of the cuent aound the loop, and you thumb gives the diection of ). Use the RHR to find the diection of τ to be up.

15 Potential Enegy of Dipole Wok must be done to change the oientation of a dipole (cuent loop) in the pesence of a magnetic field. Define a potential enegy U (with zeo at position of max toque) coesponding to this wok. x θ F θ F. U θ 90 τdθ U = θ 90 sin θdθ Theefoe, U = [ cosθ] θ 90 U cosθ = U =

16 Potential Enegy of Dipole x x x τ = 0 U = - τ = X U = 0 τ = 0 U = negative wok positive wok

17 Peflight 13: Two cuent caying loops ae oiented in a unifom magnetic field. The loops ae nealy identical, except the diection of cuent is evesed. 8) What is the toque on loop 1? a) clockwise b) counte-clockwise c) zeo Loop 1: points to the left, so the angle between and is equal to 180º, hence τ =0. 9) How does the toque on the two loops compae? a) τ 1 > τ 2 b) τ 1 = τ 2 c) τ 1 < τ 2 Loop 2: points to the ight, so the angle between and is equal to 0º, hence τ = 0. 10) Which loop occupies a potential enegy minimum, and is theefoe stable? a) loop 1 b) loop 2 c) the same U = Loop 1: U 1 = + Loop 2: U 2 = U 2 is a minimum.

18 The Hall Effect Is cuent due to motion of positive o negative chages? Positive chages moving CCW expeience upwad foce Uppe plate at highe potential Negative chages moving clockwise expeience upwad foce Uppe plate at lowe potential Equilibium between electostatic & magnetic foces: VH Fup = qvdift Fdown = qeinduced = q VH = vdift w= "Hall Voltage" w This type of expeiment led to the discovey (E. Hall, 1879) that cuent in conductos is caied by negative chages (not always so in semiconductos). Can be used as a -senso.

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