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Electricity and Magnetism Review Faraday’s Law Lana Sheridan De Anza College Dec 3, 2015 Overview • Faraday’s law • Lenz’s law • magnetic field from a moving charge • Gauss’s law Definition Reminder:(30.18) Magnetic Fluxof magnetic flux S field B that makes an s case is S B (30.19) a, then u 5 908 and the the plane as in Figure maximum value). a weber (Wb); 1 Wb 5 Magnetic flux The magnetic flux e plane is a magnetic r to the plane. S dA u S dA Figure 30.19 The magnetic flux through an area element dA S S is B ? d A 5 B dA cos u, where Sa magnetic field through of d A is a vector perpendicular to the surface. X ΦB = a surface A is B · (∆A) Units: Tm2 If the surface is a flat plane and B is uniform, that just reduces to: ΦB = B · A n a magnet is moved When the magnet is When the magnet is held d a loop of wire moved away from the Changing emf there is no ected to a sensitive flux andstationary, loop, the ammeter sh induced current in the eter, the ammeter that the induced curr theof wire there is no that a current is opposite that shown When ais magnet is atloop, rest even near when a loop potential inside the loop. ed in the loop. part a . difference across the magnet ends ofisthe wire. I I N S N b S N c S When the magnet is toward a loop of wire stationary, there is no connected to a sensitive Changing flux and emf induced current in th ammeter, the ammeter loop, eventhe when the shows a current is When the north pole of the that magnet is moved towards the loop, magnet is inside the induced in the loop. magnetic flux increases. I N .1 A simple experiment at a current is induced hen a magnet is moved away from the loop. a A current flows clockwise in the loop. S N b S the magnet is held moved away from the ary, there is no Changing flux andloop, emfthe ammeter shows d current in the that the induced current ven when the the north pole of is opposite that shown in away from the loop, When the magnet is moved a t is inside the loop. part . the magnetic flux decreases. I N S N S c A current flows counterclockwise in the loop. Faraday’s Law Faraday’s Law If a conducting loop experiences a changing magnetic flux through the area of the loop, an emf EF is induced in the loop that is directly proportional to the rate of change of the flux, ΦB with time. Faraday’s Law for a conducting loop: E=− ∆ΦB ∆t Faraday’s Law Faraday’s Law for a coil of N turns: EF = −N ∆ΦB ∆t if ΦB is the flux through a single loop. Changing Magnetic Flux The magnetic flux might change for any of several reasons: • the magnitude of B can change with time, • the area A enclosed by the loop can change with time, or • the angle θ between the field and the normal to the loop can change with time. Additional examples, video, and practice available at Lenz’s Law Lenz’s Law 30-4 Lenz’s Law An induced current has a direction such that the Faraday proposed histhe law of induc magnetic fieldSoon dueafter to the current opposes devised a rule for determining the direction of an change in the magnetic flux that induces the current. An induced current has a direction such that the ma opposes the change in the magnetic flux that induces t S The magnet's motion creates a magnetic dipole that opposes the motion. N N µ Furthermore, the direction of an induced emf is th a feel for Lenz’s law, let us apply it in two different where the north pole of a magnet is being moved to 1. Opposition to Pole Movement. The approach Basically, Lenz’s law let’s us interpret the minus Fig. 30-4 increases the magnetic flux through sign in the equation we write to represent current in the loop. From Fig. 29-21, we know t Faraday’s Law. netic dipole with a south pole and a north po : moment ! is directed from south to north. ∆Φ i B increase being E:= − caused by the approaching mag thus ! ) must ∆t face toward the approaching no S 30-4). Then the curled – straight right-hand rul Fig. 30-4 Lenz’s law at work. As the the current induced in the loop must be counte magnet is 1moved toward the loop, a current If we Figure from Halliday, Resnick, Walker, 9th ed. next pull the magnet away from th : B : 30-5b.Thus, Bind and B are now in the same direction. In Figs. 30-5c and d, the south pole of the magnet approaches and retreats from the loop, respectively. (a) (b) Lenz’s Law: Page 795 in Textbook Increasing the external field B induces a current with a field Bind that opposes the change. The induced current creates this field, trying to offset the change. The fingers are in the current's direction; the thumb is in the induced field's direction. Decreasing the external field B induces a current with a field Bind that opposes the change. Increasing the external field B induces a current with a field Bind that opposes the change. (c) Decreasing the external field B induces a current with a field Bind that opposes the change. B ind B B ind B i i i i B B B ind B ind B ind B ind B i B B ind B ind B B ind i B (a) i i i B B B ind B i i i B B ind (b) B ind (c) (d) : A ) x s o e - uniform magnetic field that exists throughout a conducting loop, with the difield B(t) perpendicular to the The rection graph givesof thethe magnitude of a uniform magnetic field that plane exists throughout a conducting directionof of the of the loop. Rankloop, the with fivethe regions field perpendicular to the plane of the loop. Rank the five regions the graph according to the magnitude of of the graph according to the magnitude of the emf induced in the emffirst. induced in the loop, greatest first. loop,the greatest Faraday’s Law Question B a b c d e t Faraday’s Law 30-4 LENZ’S LAW 795 wnward, CHECKPOINT 2 The figure shows three situations in which identical circular crease in : The figure shows three situations in whichfields identical circular conconducting loops are in uniform magnetic that are either Bind diducting loops are in uniform magnetic fields that are either inincreasing (Inc) or decreasing (Dec) in magnitude at identical then the creasing (Inc) or decreasing (Dec) in magnitude at identical rates. In each, the dashed line coincides with a diameter. Rank the ward flux rates. In each, the dashed line coincides with a diameter. Rank situations accordingaccording to the magnitude of the current induced ig. 29-21 the situations to the magnitude of the current in-in the loops, duced greatest first. 5a. in the loops, greatest first. oses the ean that he magInc Inc Dec rom the but it is Inc Dec Inc ward inn in Fig. . (a) (b) (c) gnet ap- Magnetic fields from moving charges and currents We are now moving into chapter 29. Anything with a magnet moment creates a magnetic field. This is a logical consequence of Newton’s third law. Magnetic fields from moving charges A moving charge will interact with other magnetic poles. This is because it has a magnetic field of its own. The field for a moving charge is given by the Biot-Savart Law: B= µ0 q v × r̂ 4π r 2 Magnetic fields from moving charges B= 1 Figure from rakeshkapoor.us. µ0 q v × r̂ 4π r 2 Magnetic fields from currents B= µ0 q v × r̂ 4π r 2 We can deduce from this what the magnetic field do to the current in a small piece of wire is. Current is made up of moving charges! qv = q q ∆s = ∆s = I∆s ∆t ∆t We can replace q v in the equation above. This element of current creates a Magnetic fields from currents magnetic field at P, into the page. ids ds A current-length element a differential magnetic int P. The green ! (the w) at the dot for point P : dB is directed into the i θ ˆr r P d B (into page) Current distribution This is another version of the Biot-Savart Law: Bseg = µ0 I ∆s × r̂ 4π r 2 where Bseg is the magnetic field from a small segment of wire, of length ∆s. ummation Magnetic fields fromfield currents The magnetic vector because of at any point is tangent to Magnetic field around a wire segment, viewed end-on: is a scalar, a circle. being the Wire with current into the page a currentB (29-1) that points y constant, B (29-2) f the cross The magnetic field lines produced by a current in a long straight wire Fig. 29-2 HALLIDAY REVISED Magnetic fields from currents to determine direction of the field lines (right-hand rule): GNETICHow FIELDS DUE TOthe CURRENTS es the dio a curFig. 29-2, : eld B at perpennd dion of the b) If the to the hed rathe page, i B B i (a ) The thumb is current's dire The fingers re the field vecto direction, whi tangent to a c (b ) Here is a simple right-hand rule for finding the direction of the mag set up by a current-length element, such as a section of a long wire: Right-hand rule: Grasp the element in your right hand with your extended th Magnetic field from a long straight wire The Biot-Savart Law, Bseg = µ0 I ∆s × r̂ 4π r 2 implies what the magnetic field is at a perpendicular distance R from an infinitely long straight wire: B= µ0 I 2πR (The proof requires some calculus.) Gauss’s Law for Magnetic Fields Gauss’s Law for magnetic fields.: I B · dA = 0 Where the integral is taken over a closed surface A. (This is like a sum over the flux through many small areas.) We can interpret it as an assertion that magnetic monopoles do not exist. The magnetic field has no sources or sinks. Gauss’s Law for Magnetic Fields I 32-3 INDUCED MAGNETIC FIELDS B · dA = 0 ore complicated than e does not enclose the f Fig. 32-4 encloses no ux through it is zero. only the north pole of el S. However, a south ace because magnetic like one piece of the I encloses a magnetic ttom faces and curved s B of the uniform and A and B are arbitrary s of the magnetic flux B Surface II N S Surface I PA R T 3 863 cates level of problem difficulty ILW Interactive solution is at Law for Magnetism Ch32 # 2 Flying Circus of Physics and Question, at flyingcircusofphysics.com on Gauss’s available in The The figure shows a closed surface. Along the flat top face, which has a radius of 2.0 cm, a perpendicular magnetic field B of for Magnetic Fields page. The total elect magnitude 0.30 T is directed outward. Along the flat bottom face, hrough each of five faces a die (singular E " (3.00 a magnetic flux of 0.70 of mWb is directed outward.given What by are ! the onds. What is the m " #N whereand N (" 1 to 5) is the num(a)Wb, magnitude (b) direction (inward or outward) of the magnetic flux through field that is the induced The flux is positive (outward) for N even curved part of the surface? cm and (b) 5.00 cm? or N odd. What is the flux through the sixth a closed surface. Along has a radius of 2.0 cm, a : field B of magnitude ard. Along the flat botux of 0.70 mWb is diare the (a) magnitude rd or outward) of the the curved part of the B ••8 Nonunifor 29 shows a circular cm in which an elec the plane of the pa concentric circle of (0.600 V & m/s)(r/R)t magnitude of the ind cm and (b) 5.00 cm? ••9 Uniform ele Summary • Faraday’s law • Lenz’s law • magnetic field from a moving charge • Guass’s law Homework Study!