MSTC Physics C

1. MSTC Physics C Study Guide Chapter 23 Sections 1-3

2. Electromagnetic Induction We know that a B field is produced by a current. Can a B field produce a current? Faraday says “Yes!”

3. Electromagnetic Induction • Electromagnetic induction - Using a magnetic field to bring about an emf and sometimes a current • Induced emf – a voltage brought about by changes in a B field • Changes in a B field always induce an emf……. But only induce a current if you have a closed pathway for the charge to follow.

4. Factors • Faraday investigated the factors that influence the magnitude of the induced emf: 1) time – more rapidly B field changes, the greater the induced emf 2) change of magnetic flux- greater the rate of change of Ф, the greater the induced emf

5. Magnetic Flux • Recall that magnetic flux is Ф = ∫ B • dA or BAcosΘ if B is uniform [Ф] = Tm2 = Wb

6. Faraday’s Law of Induction ε = - dФ / dt (The negative sign will be explained by Lenz’ Law)

7. Lenz’s Law • Induced emf gives rise to a current whose B field opposes the change in Ф

8. Example if magnet moves up, Ф through loop increases since B field is getting stronger therefore, I needs B field to oppose existing B field therefore (when viewed from above) gives clockwise current so B is down N S

9. Example if magnet moves down, Ф decreases through loop since B field is getting weaker therefore I needs B field to help existing B field therefore I goes cclockwise (when viewed from above) so B is up N S

10. Faraday’s Law • Since Ф = ∫ B • dA = ∫ B dAcosΘ • And ε = - dФ /dt • ε= - d/dt ∫ B dAcosΘ • So emf can be induced if: 1) B changes 2) Area of loop changes 3) orientation of loop within B field changes

11. Sample Problem • A square coil of side 5 cm contains 100 loops and is positioned perpendicular to a uniform 0.6 T B field. It is quickly and uniformly pulled from the field to a region where B drops abruptly to zero. It takes 0.1 sec for the whole coil to reach the field – free region. How much energy is dissipated in the coil if its resistance is 100 Ω?

12. Sample Problem • A plane loop of wire consisting of a single turn of cross-sectional area 8 cm2 is perpendicular to a magnetic field that increases uniformly in magnitude from 0.5 T to 2.5 T in a time of 1 s. What is the resulting induced current if the coil has a total resistance of 2 Ω?

13. Sample Problem • A plane loop of wire of 10 turns, each of area 14 cm2, is perpendicular to a magnetic field whose magnitude changes in time according to B = (0.5T)sin(60πt). What is the induced emf in the loop as a function of time?

14. Sample Problem • A long solenoid has n turns per meter and carries a current I = Io (1 – e-αt), with Io = 30 A and α = 1.6 s-1. Inside the solenoid and coaxial with it is a loop that has a radius R = 6 cm and consists of a total of N turns of fine wire. What emf is induced in the loop by the changing current? Take n = 400 turns/m and N = 250 turns.

15. Motional emf suppose B if uniform and out l of page since area of loop is increasing dФ / dt and there is an induced emf v ε = - dФ / dt = -d/dt (BAcosΘ) = -B dA/dt

16. Motional emf in time t the rod moves x = vt l so dA = ldx = lvdt and ε = -B (lvdt/dt) = - Blv v Lenz’s Law tells us that since Ф is increasing and B needs to be into the page, I must be clockwise

17. Motional emf motional emf - l voltage brought about by moving through a B field v

18. Sample Problem • A conducting rod of length l moves on two horizontal frictionless rails. If a constant force of 1 N moves the bar at 2 m/s through a magnetic field B which is into the paper, A) what is the current through an 8 Ωresistor R? B) What is the rate of energy dissipation in the resistor? C) What is the mechanical power delivered by the force F?

19. Motional emf Since dΦ/dt brings about a l current we know there is an E field that has been created to “push” the charge Recall V = ∫ E · ds so vε = ∫ E · ds Since ε = -dΦ/dt Then ∫E · ds = -dΦ/dt

20. Sample Problem • A magnetic field in an electromagnet is nearly uniform at any instant over a circular area of radius R. The current in the windings of the electromagnet is increasing in time so that B changes in time at a constant rate dB/dt at each point. Beyond the circular region (r>R) we assume B = 0. Determine the E field at any point a distance r from the center of the circular area.

21. Sample Problem • A magnetic field directed into the page changes with time according to B = (0.03t2 + 1.4) T, where t is in s. The field has a circular cross-section of radius R = 2.5 cm. What are the magnitude and direction of the electric field at point P1 when t = 3 s and r1 = 0.02 m?

22. Sample Problem • An aluminum ring of radius 5 cm and resistance 0.0003 Ωis placed on top of a long air-core solenoid with 1000 turns per meter and radius 3 cm. At the location of the ring, the magnetic field due to the current in the solenoid is one half that at the center of the solenoid. If the current in the solenoid is increasing at a rate of 270 A/s, A) what is the induced current in the ring? B) At the center of the ring, what is the magnetic field produced by the induced current in the ring? C) What is the direction of the field in B?