Last time Gauss' Law: Examples (Ampere's Law) 1
Ampere s Law in Magnetostatics iot-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, dl dl t 0I c c c ( i i ) 0 1 2 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) Ampere s Law: IC dl h nhi n windings per unit length Use Ampere s law to calculate the magnetic field inside a solenoid. (n is number of wraps per unite length). A.. C. D. 2µ ni µ ni 0 0 3µ ni 0 4µ ni 0 µ ni 0 3
Today Ampere's Law Faraday s law 4
Example: A Non-Uniform Current Distribution Long, hollow cylindrical current of current density: J cr r 2 2 Insider the cylinder, the total current encircled by the Amperian loop is r r 4 4 2 3 c( r a ) I JdA cr (2 rdr) 2 c r dr a a 2 0c 4 ( 2r) r a 2 4 0c 4r r 4 a 4 5
Example: Magnetic field of a long wire outside the wire dl 2 r 0I C 0I 2 r 6
iclicker Question 0 2 C dl r I 0 2 I r outside the wire 7 0 2 I R Assume uniform current density, what s the magnetic field vs. r inside the long wire. 0 2 2 I r R 0 2 I r 0 I r A). ). C). D).
iclicker Question outside the wire dl 2 r 0I C 0I 2 r inside the wire dl 2 r C r I R 2 0 2 0I r 2 2 R 8
Moving ar and Energy Conservation P=IV=I(emf) Are we getting something for nothing? F I F ar current I: F I Il F F I IL F m Work: W Fx ILx emf vl x Power: Main principle of electric generators: Mechanical power is converted to electric power P P P W t IL ILv I(emf ) x t 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. 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
6D-04 Earth Magnetic Field Inductor approaching moving away 12
Magnetic Flux cos A2 na 1 Wb = 1 T m 2 i n d A S Gauss s Law for Magnetism over closed surface nda 0 S NAcos (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: E dl = d dt [ nda] Sign: given by right hand rule Michael Faraday (1791-1867) Differential form of Faraday s law: E = t
Two Ways to Produce Changing emf d mag dt Two ways to produce curly electric field: 1. Changing 2. Changing A d mag d dt dt A d dt A da dt
Inductor Radio (6D-15) 16
UHF Transmitter and Dipole Receiver (6D-17) 17
Faraday s Law and Motional EMF Magnetic force approach: qe qv F tot E v emf v L I Use Faraday law: emf d mag dt A Lvt mag emf lim t0 t mag v L I
Faraday s Law and Generator emf d mag dt na ˆ wh cos t whcost emf emf d dt mag wh wh sin t d dt cos t I
Exercise A uniform time-independent magnetic field =3 T points 30 o 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 emf mag nˆ A Acos30 dt emf t 0.0264 m2 0.02 m 2 3 T 0.87 0.1 s 0.166 V
Example R emf d mag dt L I v 1 2 mag 2 A 1 A 2 1 A Lvt mag 2 1 Lvt emf mag emf vl lim vl2 1 2 1 I t 0 t 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. ε dφ where dt S nda N Minus sign indicates the sense of EMF: Lenz s Law Decide on which way n goes Fixes sign of N RHR determines the positive direction for EMF
How to use Faraday s law to determine the induced current direction n n 1. define the direction of ; can be any of the two normal direction, e.g. point to right 2. determine the sign of Φ. Here Φ>0 3. determine the sign of Φ. Here Φ >0 N 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 ds nc W q Enc ds Rewrite Faraday s Law in terms of induced electric field: E nc ds dφ dt This form relates E and! 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. 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= ZL4kbIf39s 27
iclicker Question The magnetic field is fixed, what s the direction of the induced currents in the closed rectangular loop? A. Clockwise. Counterclockwise C. No induced currents.
Example At 1, 3, and 5, is not changing. So there is no induced emf. At 2, is increasing into page. So emf is induced to produce a counterclockwise current. At 4, in decreasing into page. So current is clockwise.
iclicker Question A current directed toward the top of the page and a rectangular loop of wire lie in the plane of the page. oth 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? a. I is increasing b. I is decreasing c. I remains constant d. I is discontinuous e. Nothing can be said. I