Forces & Magnetic Dipoles BxθFF.θ μ μ = AI r r r τ = μ×B r U = −μ • B r First electric motor invented by Faraday, 1821Wire with current flow (in cup of Hg) rotates around a Bar magnet Faraday’s motor IWire with current rotates around a Permanent magnetr r r dF = I dl × B Torque on a rectangular current loop The loop will feel a torque which tends to rotate the loop, back into planeConsider a wire loop dimensions a x b whose plane is an angle φ relative to a constant B field. There will be a net torque whose magnitude on this loop is given by, rτ = IBab sin φ = IB(area )sin φ Magnetic Moment, μ , of a rectangular current loop Definition ; μ = current×area= I A μIμ is vector quantity, whose direction is normal to loop plane, use right hand rule to define direction. We can define vector torque more conveniently in terms of vector magnetic moment crossed by B fieldr τ = μ×B rrAnother name for a current loop is “magnetic dipole” r Derivation of τ = μ × B rrForces on wires of length b are collinear and cancel.Torque on a rectangular current loopr r r F = Il ×BForces on wires of length a cancel but are not collinear. There will be a torque.r r F = I l × B = IaB b τ = 2r⊥ F = 2( sin φ )( IaB) 2 r τ = IB(ab) sin φ r r r τ = μB sin φ = μ × B r r r τ = μ×Br⊥ϕϕ r = b/2F Preflight 13:r r v F = IL × BA square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. 2) What is the force on section a-b of the loop? a) zerob) out of the pagec) into the pageab: Fab = 0 = Fcd since the wire is parallel to B.3) What is the force on section b-c of the loop? a) zerob) out of the pagec) into the pagebc: Fbc = ILB RHR: I is up, B is to the right, so F points into the screen.4) What is the net force on the loop? a) zerob) out of the pager r By symmetry: F = − F da bc⇒c) into the pager r r r r F n e t = F a b + F b c + Fc d + Fd a = 0 Magnetic Moment, μ , of arbitrary loop This is generalDefinition ; μ = current×area= I A We can more generally define magnetic moments only by area and do not need to know the actual dimensions.μμμIIIFor any shape, μ = current×area= I A andr τ = AIB sinτθ = μ × B rrNote: if loop consists of N turns, μ = NAI Torque on loop in different angular positions BBτ IμμI Net torque is zero when B is parallel to μNet torque is maximum when B is perpendicular to μ Bar Magnet Analogy • You can think of a magnetic dipole moment as a bar magnet: μ=N– In a magnetic field they both experience a torque trying to line them up with the field – As you increase I of the loop Æ stronger bar magnet •We will see that such a current loop does produce magnetic fields, similar to a bar magnet. Application; galvanometer uses torques on coiled loops in magnetic field Torque is produced about the needle axis and this counter acts the restoring spring and enables the needle to rotateCurrent increased Æ μ = I • Area increases Æ Torque from B increases Æ Angle of needle increasesCurrent decreased Æ μ decreases Æ Torque from B decreases Æ Angle of needle decreasesThis is how almost all dial meters work—voltmeters, ammeters, speedometers, RPMs, etc. Example: Loop in a B-Field A circular loop has radius R = 5 cm and carries current I = 2 A in the counterclockwise direction. A magnetic field B =0.5 T exists in the negative z-direction. The loop is at an angle θ = 30° to the xy-plane. By x x x x x x x zx x x x x x xIx x x x x x xWhat is the magnetic moment μ of the loop?x x x x x x xμ = π r2 I = .0157 Am2 The direction of μ is perpendicular to the plane of the loop as in the figure. Find the x and z components of μ 😡 zμμx = –μ sin 30° = –.0079 Am2XBμz = μ cos 30° = .0136 Am2θyXx Electric Dipole Analogy +qr FEr p r Fr r r τ = r ×F r r F = qE.
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-qBxr Fr Fθ θ r μr r r τ = r ×F.r r r F = IL×B (per turn)r r p = 2qaμ = NAIr r r τ = p× Er r r τ = μ×B Preflight 13: A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right. 6) What is the net torque on the loop? a) zerob) upc) downd) out of the pagee) into the pager r r μ points out of the page (curl your fingers in the direction of the τ = μ×B current around the loop, and your thumb gives the direction of μ). Use the RHR to find the direction of τ to be up. Potential Energy of Dipole • Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field.xr F• Define a potential energy U (with zero at position of max torque) corresponding to this work. θU ≡∫ τdθ ⇒90 °Therefore,U = μB<− cos θ >θ 90°Br Fθ θ r μ.θU =∫ μB sin θdθ90 °r r ⇒ U = − μB cosθ ⇒ U = − μ • B Potential Energy of DipoleμBμxB xμx τ=0τ = μB Xτ=0U = -μBU=0U = μBnegative workpositive workB Preflight 13: Two current carrying loops are oriented in a uniform magnetic field. The loops are nearly identical, except the direction of current is reversed. 8) What is the torque on loop 1? a) clockwiseb) counter-clockwisec) zeroLoop 1: μ points to the left, so the angle between μ and B is equal to 180º, hence τ =0.9) How does the torque on the two loops compare? a) τ1 > τ2b) τ1 = τ2c) τ1 Loop 2: μ points to the right, so the angle between μ and B is equal to 0º, hence τ = 0.10) Which loop occupies a potential energy minimum, and is therefore stable? a) loop 1 b) loop 2 c) the same r r Loop 1: U = +μ B U = −μ • B 1 Loop 2: U2 = −μ B⇒ U2 is a minimum. The HallEffect• Is current due to motion of positive or negative charges?• Positive charges moving CCW experience upward force • Upper plate at higher potential• Negative charges moving clockwise experience upward force • Upper plate at lower potentialEquilibrium between electrostatic & magnetic forces: VH Fup = qvdrift B Fdown = qEinduced = q VH = vdrift Bw = “Hall Voltage” w • This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges (not always so in semiconductors). • Can be used as a B-sensor.