1. Thursday August 2, 2012 1 PHYS 1444 Ian Howley PHYS 1444 Lecture #15 Thursday August 2, 2012 Ian Howley Dr. B will assign final (?) HW today(?) It is due next Thursday(?) Inductance L/LR/LRC Circuits

2. Thursday August 2, 2012 Self Inductance The concept of inductance applies to a single isolated coil of N turns. How does this happen? –When a changing current passes through a coil –A changing magnetic flux is produced inside the coil –The changing magnetic flux in turn induces an emf in the same coil –This emf opposes the change in flux. Whose law is this? Lenz’s law What would this do? –When the current through the coil is increasing? The increasing magnetic flux induces an emf that opposes the original current This tends to impede the increased current –When the current through the coil is decreasing? The decreasing flux induces an emf in the same direction as the current 2 PHYS 1444 Ian Howley

3. Thursday August 2, 2012 Self Inductance Since the magnetic flux  B passing through an N turn coil is proportional to current I in the coil, We define self-inductance, L : Self Inductance What does magnitude of L depend on? –Geometry and the presence of a ferromagnetic material Self inductance can be defined for any circuit or part of a circuit –What is the unit for self-inductance? The induced emf in a coil of self-inductance L is – 3 PHYS 1444 Ian Howley

4. Thursday August 2, 2012 Inductor An electrical circuit always contains some inductance but it is often negligible –If a circuit contains a coil of many turns, it could have a large inductance A coil that has significant inductance, L, is called an inductor and is express with the symbol –Precision resistors are normally wire wound Would have both resistance and inductance The inductance can be minimized by winding the wire back on itself in opposite direction to cancel magnetic flux This is called a “non-inductive winding” For an AC current, the greater the inductance the less the AC current –An inductor thus acts like a resistor to impede the flow of alternating current (not to DC, though. Why?) –The quality of an inductor is indicated by the term reactance or impedance (see section 30-7) 4 PHYS 1444 Ian Howley

5. Energy Stored in a Magnetic Field The work done to the system is the same as the energy stored in the inductor when it is carrying current I – Energy Stored in a magnetic field inside an inductor –This is compared to the energy stored in a capacitor, C, when the potential difference across it is V –Just like the energy stored in a capacitor is considered to reside in the electric field between its plates –The energy in an inductor can be considered to be stored in its magnetic field Thursday August 2, 2012 5 PHYS 1444 Ian Howley

6. Thursday August 2, 2012 LR Circuits This can be shown w/ Kirchhoff rule loop rules –The emfs in the circuit are the battery voltage V 0 and the emf  =- L (d I /dt) (opposes the  I after the bat. switched on) –The sum of the potential changes through the circuit is –Where I is the current at any instance –By rearranging the terms, we obtain a differential eq. – –We can integrate just as in RC circuit –So the solution is –Where  =L/R This is the time constant  of the LR circuit and is the time required for the current I to reach 0.63 of the maximum 6 PHYS 1444 Ian Howley

7. Discharge of LR Circuits If the switch is flipped away from the battery –The differential equation becomes – –So the integration is –Which results in the solution – –The current decays exponentially to zero with the time constant  =L/R –So there always is a reaction time when a system with a coil, such as an electromagnet, is turned on or off. –The current in LR circuit behaves in a similar manner as for the RC circuit, except that in steady state RC current is 0, and the time constant is inversely proportional to R in LR circuit unlike the RC circuit 7

8. 30-5 LC Circuits and Electromagnetic Oscillations An LC circuit is a charged capacitor shorted through an inductor.

9. 30-5 LC Circuits and Electromagnetic Oscillations Summing the potential drops around the circuit gives a differential equation for Q: This is the equation for simple harmonic motion, and has solutions..

10. 30-5 LC Circuits and Electromagnetic Oscillations Substituting shows that the equation can only be true for all times if the frequency is given by The current is sinusoidal as well:

11. 30-5 LC Circuits and Electromagnetic Oscillations The charge and current are both sinusoidal, but with different phases.

12. 30-5 LC Circuits and Electromagnetic Oscillations The total energy in the circuit is constant; it oscillates between the capacitor and the inductor:

13. 30-6 LC Oscillations with Resistance ( LRC Circuit) Any real (nonsuperconducting) circuit will have resistance.

14. 30-6 LC Oscillations with Resistance ( LRC Circuit) Now the voltage drops around the circuit give The solutions to this equation are damped harmonic oscillations. The system will be underdamped for R 2 4L/C. Critical damping will occur when R 2 = 4L/C.

15. 30-6 LC Oscillations with Resistance ( LRC Circuit) This figure shows the three cases of underdamping, overdamping, and critical damping.

16. 30-6 LC Oscillations with Resistance ( LRC Circuit) The angular frequency for underdamped oscillations is given by The charge in the circuit as a function of time is..