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Lecture 17-1 Ampere’s Law in Magnetostatics Biot-Savart’s Law can be used to derive another relation: Ampere’s Law The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, C Bt dl 0 (i1 i2 ) C B d l 0 I C • Choosing a direction of integration. • A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration. Lecture 17-2 Magnetization and “Bound” Current in Matter • Strong externally applied field Bapp aligns the magnetic moments in matter. Magnetization M V d M dV Magnetic susceptibility M B app Bm m Bapp m M B app 0 B Bapp 0 M (1 m ) Bapp Relative permeability Km B K m B app B app 0 K m 0 permeability Lecture 17-3 Hysteresis for a Ferromagnet Lack of retraceability shown is called hysteresis. Memory in magnetic disk and tape Alignment of magnetic domains retained in rock (cf. lodestones) Area enclosed in hysteresis loop Energy loss per unit volume hard magnet: broad hysteresis loop (hard to demagnetize, large energy loss, highe memory) soft magnet: narrow hysteresis loop (easy to demagnetize,…) Lecture 17-4 Magnetic Flux B B BA2 cos B nA 1 Wb = 1 T m2 B B ndA S Bi Gauss’s Law for Magnetism S B ndA 0 over closed surface B NBA cos (N turns) Lecture 17-5 Faraday’s Law of Induction The magnitude of the induced EMF in conducting loop is equal to the rate at which the magnetic flux through the surface spanned by the loop changes with time. dΦB ε dt where B S B ndA N Minus sign indicates the sense of EMF: Lenz’s Law • Decide on which way n goes Fixes sign of B • RHR determines the positive direction for EMF N Lecture 17-6How to use Faraday’s law to determine the induced current direction 1. define the direction of n ; can be any of the two normal direction, e.g. n point to right 2. determine the sign of Φ. Here Φ>0 N 3. determine the sign of ∆Φ. Here ∆Φ >0 4. determine the sign of using faraday’s law. Here <0 5. RHR determines the positive direction for EMF • If >0, current follow the direction of the curled fingers. • If <0, current goes to the opposite direction of the curled fingers. Lecture 17-7 Conducting Loop in a Changing Magnetic Field Induced EMF has a direction such that it opposes the change in magnetic flux that produced it. approaching Magnetic moment created by induced currrent I repels the bar magnet. Force on ring is repulsive. moving away Magnetic moment created by induced currrent I attracts the bar magnet. Force on ring is attractive. Lecture 17-8 Induced Electric Field from Faraday’s Law • EMF is work done per unit charge: ε W /q • If work is done on charge q, electric field E must be present: ε E nc W q Enc ds ds Rewrite Faraday’s Law in terms of induced electric field: E nc dΦB ds dt This form relates E and B! B • Note that E ds 0for E fields generated by charges at rest (electrostatics) since this would correspond to the potential difference between a point and itself. => Static E is conservative. • The induced E by magnetic flux changes is non-conservative. Lecture 17-9 Warm-up quiz The magnetic field is decreasing, what’s the direction of the induced currents in the closed rectangular loop? A. Clockwise B. Counterclockwise C. No induced currents. Lecture 17-10 Faraday’s and Lenz’s Laws At 1, 3, and 5, B is not changing. So there is no induced emf. At 4, B in decreasing into page. So current is At 2, B is increasing into page. So clockwise. emf is induced to produce a counterclockwise current. Lecture 17-11 Motional EMF of Sliding Conductor Induced EMF: Lenz’s Law gives direction counter-clockwise Faraday’s Law FM decelerates the bar dv B 2l 2 v m dt R dv B 2l 2 dt v mR v(t ) v 0 e B 2l 2 t mR dB dx Bl Blv dt dt This EMF induces current I Blv I R R Magnetic force FM acts on this I B 2l 2 v FM I lB R Lecture 17-12 Ways to Change Magnetic Flux B BA cos • Changing the magnitude of the field within a conducting loop (or coil). • Changing the area of the loop (or coil) that lies within the magnetic field. • Changing the relative orientation of the field and the loop. motor generator Lecture 17-13 Other Examples of Induction + - Switch has been open for some time: Switch is just closed: Nothing happening EMF induced in Coil 2 + - Switch is just opened: EMF is induced again Switch is just closed: EMF is induced in coil - + Back emf (counter emf) Lecture 17-14 PHYS241 - Quiz A A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. Both are held in place by an external force. If the current I is decreasing, what is the direction of the magnetic force on the left edge of the loop? a. Toward the right b. Toward the left c. Toward top of page d. Toward bottom of page e. No force acts on it. I Lecture 17-15 PHYS241 - Quiz B A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. If the current I is increasing, what happens to the loop? a. The loop is pulled toward the top of the page b. The loop is pulled toward the current c. A clockwise current is induced in the loop. d. A counterclockwise current is induced in the loop. e. Nothing happens to the loop I Lecture 17-16 PHYS241 - Quiz C A current directed toward the top of the page and a circular loop of wire lie in the plane of the page. If a clockwise current is induced in the loop by the current I, what can you conclude about it? I a. I is increasing b. I is decreasing c. I remains constant d. I is discontinuous e. Nothing can be said.