General Review Electrostatics Magnetostatics Electrodynamics motion of “q” in external E-field E-field generated by qi Magnetostatics motion of “q” and “I” in external B-field B-field generated by “I” Electrodynamics time dependent B-field generates E-field ac circuits, inductors, transformers, etc time dependent E-field generates B-field

LECTURE 16 Faraday’s Law of Induction dA B

Induction: DEMONSTRATION *Current flows only if there is relative motion between the loop and the magnet *Current disappears when the relative motion ceases *Faster motion produces a greater current DEMOS: 6D-04 – Earth’s magnetic field – hoop and bar magnetic Caution – this picture is not an example of right hand rule! ammeter 11/7/2018

Induction Effects Bar magnet moves through coil: Current induced in coil v S N Change pole that enters: Induced current changes sign v N S Bar magnet stationary inside coil: No current induced in coil DEMO: 6D-04 Earth’s Magnet Field – hoop and bar magnet N S Coil moves past fixed bar magnet: Current induced in coil v S N Change direction of motion in any of above examples  Induced current changes sign 11/7/2018

Induction Effects from Currents www.daviddarling.info/images/electromagnetic_induction When the switch is closed (or opened) current induced in coil b Steady state current in coil a no current induced in coil b DEMO: 6D- 01 & 6D-02 Conclusion: A current is induced in a loop when: there is a change in magnetic field through it. This can happen many different ways. How can we quantify this?

A wire loop falling into a B field (increasing) Induction Example A wire loop falling into a B field (increasing) downward Velocity v magnetic Field B Force acting on moving charges time N N N 11/7/2018

Faraday’s Law An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing. Only components perpendicular to loop dA B Define the flux of the magnetic field through an open surface A as: ITEMS IN RED BOLD ARE FOR WRITING/DiSCUSSING IN CLASS. REMOVE FROM POSTED NOTES Unit: 1 Weber=1Wb=1 Tm2 11/7/2018

Faraday’s Law Restated The magnitude of the emf  induced in a conducting loop is equal to the rate at which the magnetic flux M through the loop changes. dA B The minus sign indicates opposition 11/7/2018

EMF for a Coil of N turns 11/7/2018

How to Change Magnetic Flux in a Coil DEMO: 6D-06 Search Coil 11/7/2018

Lenz’ Law (to determine direction of induced I in loop) An induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current. Opposition to Flux: B induced always opposes the change in the flux of B, but does not always point opposite it!!! v B S N *Move N towards loop. FluxB increases. Induced I sets up its own field to oppose the increase in FluxB. I must be counterclockwise. Bi *Move N away from loop. FluxB decreases. Induced I sets up its own field to oppose the decrease in FluxB. I must be clockwise. v B S N Direction of I must be to oppose change...otherwise violate conservation of energy. 11/7/2018

Lenz’ Law An induced current has a direction such that the magnetic field due to the induced current opposes the change in the magnetic flux that induces the current. Opposition to Flux: v B S N v B S N 11/7/2018

What is the direction of the induced current in the loop? Example 1 X X X X X X X X X X X X v x y A conducting rectangular loop moves with constant velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown. What is the direction of the induced current in the loop? No induced current. B is constant, A is constant, v is constant. dFluxB/Dt=0 (a) ccw (b) cw (c) no induced current 11/7/2018

Example 2 A conducting rectangular loop moves with constant velocity v in the -y direction and a constant current I flows in the +x direction as shown. What is the direction of the induced current in the loop? v I x y Cw=clockwise As loop moves in -y direction, B decreases from I in wire. B from I wire points into the page. To oppose the decrease IN FLUX FROM THE DECREASE IN B, Binduced will point into the page (to try to increase the flux) & current will be clockwise (a) ccw (b) cw (c) no induced current 11/7/2018

Demo: E&M Cannon Connect solenoid to a source of alternating voltage. ~ side view v B Connect solenoid to a source of alternating voltage. The flux through the area ^ to axis of solenoid therefore changes in time A conducting ring placed on top of the solenoid will have a current induced in it opposing this change. There will then be a force on the ring since it contains a current which is circulating in the presence of a magnetic field. Note that it’s the off-axis component of B (the “fringe field”) that flings the ring. DEMO: 6D-11 Jumping Ring – EM Cannon 11/7/2018

Example 4 Suppose two identically shaped rings are used in the demo. Ring 1 is made of copper (resistivity = 1.7X10-8 m); Ring 2 is made of aluminum (resistivity = 2.8X10-8 m). Let F1 be the force on Ring 1; F2 be the force on Ring 2. Ring 1 Ring 2 (a) F2 < F1 (b) F2 = F1 (c) F2 > F1 11/7/2018

Faraday’s Law in terms of E Field x x x x x x x x x x r E B 11/7/2018

Example 5 t B z 1 X X X X X X X X X X X X X X X X X X X X X X X X X X X X x y The magnetic field in a region of space of radius 2R is aligned with the –z direction and changes in time as shown in the plot. What is sign [direction] of the induced emf in a ring of radius R at time t=t1? (a)  < 0 (E cw) (b)  = 0 (c)  > 0 (E ccw) 11/7/2018

Example 6 t B z 1 X X X X X X X X X X X X X X X X X X X X X X X X X X X X x y The magnetic field in a region of space of radius 2R is aligned with the –z direction (into the page) and changes in time as shown in the plot. What is the relation between the magnitudes of the induced electric fields ER at radius R and E2R at radius 2R? 2R R (a) E2R = ER (b) E2R = 2ER (c) E2R = 4ER 11/7/2018

Faraday’s Law in terms of Electric Fields Summary Faraday’s Law (Lenz’ Law) a changing magnetic flux through a loop induces a current in that loop negative sign indicates that the induced EMF opposes the change in flux Faraday’s Law in terms of Electric Fields 11/7/2018

I2 is decreasing in magnitude I2 is constant QUIZ lecture 16 Two current loops are perpendicular to the z axis and are centered on the this axis. Current I2 is clockwise. I1 is the induced current in the upper loop. If I1 is counterclockwise, which statement is true? z I2 is decreasing in magnitude I2 is constant I2 is increasing in magnitude I1 y x I2 11/7/2018

I2 is decreasing in magnitude I2 is constant QUIZ lecture 16 Two current loops are perpendicular to the z axis and are centered on the this axis. Current I2 is counterclockwise. I1 is the induced current in the upper loop. If I1 is counterclockwise, which statement is true? z I2 is decreasing in magnitude I2 is constant I2 is increasing in magnitude I1 y x I2 11/7/2018

I1 is decreasing in magnitude I1 is constant QUIZ lecture 16 Two current loops are perpendicular to the z axis and are centered on the this axis. Current I1 is counterclockwise. I2 is the induced current in the bottom loop. If I2 is clockwise, which statement is true? z I1 is decreasing in magnitude I1 is constant I1 is increasing in magnitude I1 y x I2 11/7/2018

I1 is decreasing in magnitude I1 is constant QUIZ lecture 16 Two current loops are perpendicular to the z axis and are centered on the this axis. Current I1 is clockwise. I2 is the induced current in the bottom loop. If I2 is clockwise, which statement is true? I1 is decreasing in magnitude I1 is constant I1 is increasing in magnitude 11/7/2018

QUIZ lecture 16 Suppose a loop is placed in a uniform magnetic field and is spun around a vertical axis as shown. The loop makes one complete revolution every second. The current induced in the loop is zero changes direction once per second changes direction twice per second 11/7/2018