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Last time Gauss' Law: Examples (Ampere's Law) 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, B dl B dl I c t 0 (i1 i2 ) c 0 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. 2 iClicker question (last class) Use Ampere’s law to calculate the magnetic field inside a solenoid. (n is number of wraps per unite length). A. B. • Ampere’s Law: C. B dl Bh I C nhI n windings per unit length B μ0nI D. B 2 μ0 nI B μ0nI B 3 μ0 nI B 4 μ0 nI 3 Today Ampere's Law Faraday’s law 4 Example: A Non-Uniform Current Distribution Long, hollow cylindrical current of current density: J cr 2 r 2 Insider the cylinder, the total current encircled by the Amperian loop is r r a a I JdA cr 2 (2 rdr ) 2 c r 3dr c( r 4 a 4 ) 2 B(2r ) B 0c 2 r 4 a 4 r 4 a4 4r 0c 5 Example: Magnetic field of a long wire outside the wire C B d l 2 rB 0 I I B 0 2 r 6 iClicker Question outside the wire C B d l 2 rB 0 I I B 0 2 r Assume uniform current density, what’s the magnetic field vs. r inside the long wire. A). 0 I B 2 R 0 I B). B r 2 2 R C). 0 I B 2 r D). 0 I B r 7 iClicker Question outside the wire C B d l 2 rB 0 I I B 0 2 r inside the wire C B d l 2 rB r2 0 I R2 0 I B r 2 2 R 8 Moving Bar and Energy Conservation P=IV=I(emf) Are we getting something for nothing? Bar – current I: FI F Fm FI = IDl ´ B = -F FI = ILB Work: emf = vBL x W = FDx = ILBDx W Dx = ILB Power: P = Dt Dt Main principle of electric generators: Mechanical power is converted to electric power P = ILBv P = I (emf ) 9 Right now, can you answer the following questions? 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. Faraday’s Law: Electromagnetic Induction • We have seen that an electric current produces a magnetic field. Can magnetic fields produce electric currents? • An electric field is produced when there is a changing magnetic field. • In a closed electric circuit, that means current is generated due to the changing magnetic field. 11 approaching 6D-04 Earth Magnetic Field Inductor moving away 12 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) Faraday’s Law emf d mag dt Faraday’s law cannot be derived from the other fundamental principles we have studied Formal version of Faraday’s law: 𝑑 𝐸 ∙ 𝑑𝑙 = − [ 𝐵 ∙ 𝑛𝑑𝐴] 𝑑𝑡 Sign: given by right hand rule Michael Faraday (1791 - 1867) Differential form of Faraday’s law: 𝜕𝐵 𝛻×𝐸 =− 𝜕𝑡 Two Ways to Produce Changing emf d mag dt Two ways to produce curly electric field: 1. Changing B 2. Changing A d mag dt d dB dA B A A B dt dt dt Inductor Radio (6D-15) 16 UHF Transmitter and Dipole Receiver (6D-17) 17 Faraday’s Law and Motional EMF ‘Magnetic force’ approach: Ftot qE qv B E vB emf vB L I Use Faraday law: emf d mag dt mag B A B Lvt emf lim t 0 mag t vB L I Faraday’s Law and Generator emf d mag dt B nˆA Bwh cos t Bwh cos t emf d mag dt d Bwh cost dt emf Bwh sin t I Exercise A uniform time-independent magnetic field B=3 T points 30o to the normal of the rectangular loop. The loop moves at constant speed v 1. What is the emf? 2. In 0.1 s the loop is stretched to be 0.12 m by 0.22 m. What is average emf during this time? d mag emf B nˆA BAcos 30 dt emf t 3 T0.0264 m 2 0.02 m 2 0.87 0.1 s 0.166 V Example R L emf d mag I B1 dt v B2 mag B2 A B1A B2 B1 A mag B2 B1 Lvt emf lim t 0 mag t Lvt vLB2 B1 I emf vLB2 B1 R R Faraday’s Law of Induction (More Quantitative) 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 How 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. 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. 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. iClicker Question 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. 6D-11 Jumping Ring Is there any differences in the two rings ? Why one can jump up, the other can’t ? http://www.youtube.com/watch?v= ZL4kbBIf39s 27 iClicker Question The magnetic field is fixed, what’s the direction of the induced currents in the closed rectangular loop? A. Clockwise B. Counterclockwise C. No induced currents. Example 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. iClicker Question 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 iClicker Question 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.