M Agnetic Fields • Determination of the Magnetic Dipole Moment of A

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303 œ Physics for Engineers II - Laboratory Experiment # 3: Determination of the magnetic moment of a magnet. Charge to mass ratio of the electron

• M agnetic fields • Determination of the magnetic dipole moment of a magnet: harmonic oscillationsof a permanent magnet in a uniform magnetic field • Charge to mass ratio of the electron: electron beam bent by a magnetic field

1 303 œ Physics for Engineers II - Laboratory Magnetic fields (I)

Sources of magnetic fields: moving charges or currents

Units of magnetic field: SI, T (Tesla), Gauss is also used(1 T = 104 G)

Biot-Savart law: field for a wire carrying a current i C C C µ “ i(dl × R) B = o ” 4π ‘ R3 C

µ0, permeability of free space µ = 4π 10-7 N/A2 0 2 303 œ Physics for Engineers II - Laboratory Magnetic fields (II)

2 Magnetic field for a µo a i Bth = 3 circular loop: 2 (x2 + a 2 ) 2 2 µo N a i Magnetic fieldfor N loops: Bth = 3 2 (x2 + a 2 ) 2

Helmholtzcoils(the distance between the two coilsis equalto the radiusof the coils) À ¤ Œ Œ µ Nia2 Œ 1 1 Œ B = o Ã + ‹ th 2 3 3 Œ» 2 2 ÿ 2 » 2 2 ÿ 2 Œ Œ (x − a ) + a (x + a ) + a Œ Õ … 2 ⁄Ÿ … 2 ⁄Ÿ ›

µo N i 8 x = 0 Bth = 3 a 5 2 3 303 œ Physics for Engineers II - Laboratory Magnetic moment of a magnet (I)

C C C τ = µ × B τ = µBsinθ

d 2θ I =τ dt 2

d 2θ µB small oscillations I + µBθ = 0 ω = dt2 I

4 303 œ Physics for Engineers II - Laboratory Magnetic moment of a magnet (II)

1/ 2 3/ 2 2 2π ≈ I ’ 2 5 (2π ) I a 1 T = = 2π ∆ ÷ T = ω « µ B ◊ 8µ0N µ i µo N i 8 B = 3 a 5 2 I a 1 T 2 = cons. µ i 1 y x I = M(3r 2 + L2 ) slope 12 5 303 œ Physics for Engineers II - Laboratory Charge to mass ratio of the electron C C Fe = qE

C C C mv2 e F = q(v × B) = evB v = rB m r m

1/ 2 1 ≈ U ’ mv2 = eU v = ∆2e ÷ 2 « m ◊

e U = 2 2 2 m r B 6 303 œ Physics for Engineers II - Laboratory Equipment

MMAAGGNNEETTIICC MMOOMMEENNTT ooff aa MMAAGGNNEETT -- HHeellmmhhoollttzz ccooiillss -- ((TTeessllaammeetteerr)) HANDLE W ITH CARE! -- MMuulliimmeetteerr ((AAmmmmeetteerr,, rraannggee AA)) -- PPoowweerr ssuuppppllyy -- CCyylliinnddrriiccaall mmaaggnneett HANDLE W ITH EXTREME CARE! (checkdimensions!!!) -- SSttooppwwaattcchh -- CCoonnnneeccttiinngg ccoorrddss -- SSuuppppoorrtt bbaassee aanndd rrooddss,, aanndd tthhrreeaadd 7 303 œ Physics for Engineers II - Laboratory Experimental procedure (I)

M easuringmagneticfields: Use a teslameter to check the value and uniformity of the magnetic fieldin the central region of Helmholtzcoils.

1. Teslameter calibration 2. Circuit set-up with Helmholtz coils and ammeter 3. Measurement of the magnetic fieldin the centre of the coils (average of the measurements of the two sides)

To hooks 1 and 2

8 303 œ Physics for Engineers II - Laboratory Experimental procedure (II)

Determination of the magnetic moment of a permanent magnet: 1. Hang a magnet ona thread tiedto its central point, so that it oscillates in a horizontal plan, in the central region of the Helmholtz coils. 2. Calculate the moment of inertia of the magnet andits uncertainty: M = (77.60 ± 0.10) 10-3 Kg, r = (5.00 ± 0.05) 10-3 m, i L = (13.90 ± 0.10) 10-2 m 3. Make the magnet oscillate: shift it from its equilibrium position, and measure the periodof oscillations for different currents (periodof 20 oscillations)

i (A) t (s) T(s) 1/i (A-1) ? is obtained from a least-squares fit 0.75 − − − 1 − − − 53/ 2 (2π )2 I a 1 1.25 − − − T 2 = ... − − − 8µ N µ i 0 9 ... − − − 303 œ Physics for Engineers II - Laboratory Results

1. Test of the uniformityof a magnetic fieldin the central regionof Helmholtzcoils. 2. Determination of the moment of inertia of the magnet and its uncertainty. 3. Record of the oscillation periods for different currents, least-squares fit, andcalculation of the magnetic moment of the magnet andits uncertainty. 4. Charge to mass ratio of the electron: Measurement of intensity(andmagnetic fieldcreated), andof accelerating voltage. Calculation of the charge to mass ratio and contrast with data in literature. 10