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Lecture 22 current loops. sources of magnetic field.

- 1.Lecture 22 Current loops. Sources of magnetic field.

2. Force on a square loop of current The square loop below has side length L and carries a current I. The magnetic field B points out of the screen and is uniform. What is the net force on the loop?F B IFFFMagnitude of force is the same on all four sides: F = ILBNet force is zero. 3. Current loop in a uniform B field r B(B in any direction, not necessarily in the plane of the screen) dl r F =rr B Idl=Ir( ) dlr Br=0=0r Constant B and IThe net force on a closed loop of current in a uniform B field is always zero. 4. Torque on a square loop of current The square loop below has side length L and carries a current I. The magnetic field B is uniform. Net force = 0 F =0BFNet torque about the center of the loop: =F= FLF= IL2BI F =0L L +F 2 2F = ILB 5. Torque on a square loop of current (2) Now the loop is in the xy plane and B is parallel to the xz plane.BFLI zFRNear side has force FN = ILB cos out of the screen. Far side has force FF = ILB cos into of the screen. These forces cancel out and dont do torque.xFL = FR = ILBNet torque about the center of the loop: L r L = FL sin + FR sin j = IL2B sin j 22 6. Magnetic dipole For a current loop of area A and current I:r r = IAMagnetic dipole moment of a current loop (= magnetic dipole)r A = area vector, with direction given by the right-hand rule Then, the torque by the uniform magnetic field is:r r r 2 = IL B sin j = B sin j = Br r r = BB Izx 7. Work by this torque as loop plane moves from 1 to 2:W =212 2 r r d = d = B sin d = + B ( cos 2 cos 1 ) 11r is clockwise r is counterclockwiser r r BMotion of a magnetic dipole (current loop) in a uniform B field given by:W = Ur r U = B cos = B Minimum (stable equilibrium) at = 0r r r = B r r U = B tends to align itself with BCurrent loop in magnetic field 8. ACT: Two turns A cable forms a circular circuit of radius R. When connected to a battery, current flows through it and we can assign it a magnetic moment . If we use the same cable to make a circular circuit with two turns of radius R/2, and use the same battery, the magnetic moment is: A. B. /2Rule of thumb: if there are N turns, count area as NA (A = area of one loop)C. 2I I I = I R 2equivalent to2I2R 1 1 = 2I = I R 2 = 2 2 2 9. MRI (Magnetic Resonance Imaging) and NMR (Nuclear Magnetic Resonance) A single proton (like the one in every hydrogen atom) has a charge (+|e|) and an intrinsic angular momentum (spin). If we (naively) imagine the charge circulating in a loop magnetic dipole moment . In an external B-field: Classically: there will be torque unless is aligned along B or against it. Quantum Mechanics: The spin is always ~aligned along B or against it Aligned: U1 = BAnti-aligned: U2= BU U2 U1 = 2B = 2.82 10 26 J proton = 1.411026 Am2 B = 1 Tesla (= 104 Gauss) Big field!In QM, you will learn that photon energy = frequency Plancks constant h 6.63 10-34 J s2.82 10 26 J f = = 42.5 MHz 6.63 10 34J s 10. If we bathe the protons in radio waves at this frequency, the protons can flip back and forth. If we detect this flipping hydrogen! The presence of other molecules can partially shield the applied B, thus changing the resonant frequency (chemical shift). Looking at what the resonant frequency is nearby. what molecules areFinally, because f U B , if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution.BSmall B low freq.Bigger B high freq. Signal at the right frequency only from this slice! 11. Thanks to 12. What produces magnetic fields? A moving charge experiences a force in a B-field By symmetry, it is reasonable to think that B fields are also produced by moving chargesgenerates B-field and exerts a force on moving charge 1moving charge 2 generates B-field and exerts a force on 13. Magnetic field by a moving charge: experimental facts When q larger, and when v larger, larger B-field produced 1 B-field decreases with 1/distance2 from the moving source B 2 r B-field is NOT directed away or towards moving chargecharge (q > 0) moving into screenB-field line is circular around moving charge 14. Magnetic field by a moving charge: equation r vr r B-field at point P B = 0 q 4 r2 r = unit vector from charge to P0 permeability constant = 4 10 7 T m/AIf q > 0 then r r B same direction as v r