1. Physics 122B Electricity and Magnetism
Lecture 23 (Knight: )
Inductance and LR Circuits
Martin Savage

2. Lecture 23 Announcements
Lecture HW has been posted and is due on Wednesday at 10 PM.
Midterm examination 3 is this coming Friday---questions from lecture, tutorial AND Lab.
10/9/2018
Physics 122B - Lecture 23

3. Generators
The figure shows a coil with N turns rotating in a magnetic field, with the coil connected to an external circuit by slip rings that transmit current independent of rotation. The flux through the coil is:
Therefore, the device produces emf and current that will vary sinusoidally, alternately positive and negative. This is called an alternating current generator, producing what we call AC voltage.
10/9/2018
Physics 122B - Lecture 23

4. Example: An AC Generator
A coil with area 2.0 m2 rotates in a 0.10 T magnetic field at a frequency of 60 Hz. How many turns are needed to generate an AC emf with a peak voltage of 160 V?
10/9/2018
Physics 122B - Lecture 23

5. Transformers
When a coil wound around an iron core is driven by an AC voltage V1cos wt, it produces an oscillating magnetic field that will induce an emf V2cos wt in a secondary coil wound on the same core. This is called a transformer.
The input emf V1 induces a current I1 in the primary coil that is proportional to 1/N1. The flux in the iron is proportional to this, and it induces an emf V2 in the secondary coil that is proportional to N2. Therefore, V2 = V1(N2/N1). From conservation of energy, assuming no losses in the core, V1I1 = V2I2. Therefore, the currents in the primary and secondary are related by the relation I1 = I2(N2/N1).
A transformer with N2>>N1 is called a step-up transformer, which boosts the secondary voltage. A transformer with N2<<N1 is called a step-down transformer, and it drops the secondary voltage.
10/9/2018
Physics 122B - Lecture 23

6. The Tesla Coil
A special case of a step-up transformer is the Tesla coil. It uses no magnetic material, but has a very high N2/N1 ratio and uses high-frequency electrical current to induce very high voltages and very high frequencies in the secondary.
There is a phenomenon called “the skin effect” that causes high frequency AC currents to reside mainly on the outer surfaces of conductors. Because of the skin effect, one does not feel (much) the electrical discharges from a Tesla coil.
10/9/2018
Physics 122B - Lecture 23

7. Metal Detectors
Metal detectors like those used at airports can detect any metal objects, not just magnetic materials like iron. They operate by induced currents.
A transmitter coil sends high frequency alternating currents that will induce current flow in conductors in its field. Because of Lenz’s Law, the induced current opposes the field from the transmitter, so that net field is reduced. A receiver coil detects the reduction in the magnetic fields from the transmitter and registers the presence of metal.
10/9/2018
Physics 122B - Lecture 23

8. (Self-) Inductance
We define the inductance L of a coil of wire producing flux Fm as:
The unit of inductance is the henry: 1 henry = 1 H = 1 T m2/A = 1 Wb/A
The circuit diagram symbol used to represent inductance is:
Example: The inductance of a long solenoid with N turns of cross sectional area A and length l is:
10/9/2018
Physics 122B - Lecture 23

9. Example: Length of an Inductor
An inductor is made by tightly winding 0.30 mm diameter wire around a 4.0 mm diameter cylinder.
What length cylinder has an inductance of 10 mH?
10/9/2018
Physics 122B - Lecture 23

10. Potential Across an Inductor
10/9/2018
Physics 122B - Lecture 23

11. Potential Across an Inductor (2)
10/9/2018
Physics 122B - Lecture 23

12. Inductors and Sparks
An inductor responds to an interruption in current flow by developing a very large potential (L dI/dt) across its terminals. This potential is typically dropped across the object (usually a switch) that is effecting the change in current. Therefore, breaking a closed circuit with an inductor often results in a spark across the switch contacts. Over time, this can erode and destroy the switch.
Spark plugs in automobiles are driven by inductances that are electronically interrupted.
10/9/2018
Physics 122B - Lecture 23

13. Example: Large Voltage across an Inductor
A 1.0 A current passes through a 10 mH inductor coil.
What potential difference is induced across the coil if the current drops to zero in 5 ms?
( 1.0 A )
10/9/2018
Physics 122B - Lecture 23

14. Question The potential at a is higher than the potential at b.
Which of the following statements about the inductor current I is consistent with this observation?
I is from a to b and steady.
I is from a to b and increasing
I is from a to b and decreasing
I is from b to a and steady
I is from b to a and increasing
10/9/2018
Physics 122B - Lecture 23

15. Energy in Inductors and Magnetic Fields
A magnetic field stores considerable energy. Therefore, an inductor, which operates by creating a magnetic field, stores energy. Let’s consider how much energy UL is stored in an inductor L carrying current I:
A solenoid of length l , area A, and N turns has L = m0N2A/l, so:
10/9/2018
Physics 122B - Lecture 23

16. Energy Density in the Fields
This is the magnetic analog of the energy stored in an electric field 2
UE = e E
Magnetic
Electric
10/9/2018
Physics 122B - Lecture 23

17. The LR Circuit
10/9/2018
Physics 122B - Lecture 23

18. Example: Exponential Decay in an LR Circuit
The switch shown has been in position a for a long time. It changes to position b at t=0.
a. What is the current in the circuit at t = 5 ms?
b. At what time has the current decayed to 1% of its original value?
10/9/2018
Physics 122B - Lecture 23

19. Plumber’s RL Analogy Valve Constriction P1 P2 Pump Flywheel
The “plumber’s analogy” of an RL circuit is a pump (=battery) pumping water in a closed loop of pipe that includes a valve (=switch), a constriction (=resistor), and rotating flywheel turned by the flow. When the valve starts the flow, the flywheel begins to spin until a steady flow is achieved and the pressure difference across the flywheel (P2-P3) goes to zero. P3
Pump = Battery
Valve = Switch
Constriction = Resistor
Flywheel = Inductor
Pressure = Potential Water Flow = Current
10/9/2018
Physics 122B - Lecture 23

20. Capacitors: Early and Late*
Initially, when a switch closes there is a potential difference of 0 across an uncharged capacitor. After a long time, the capacitor reaches its maximum charge and there is no current flow through the capacitor. Therefore, at t=0 the capacitor behaves like a short circuit (R=0), and at t=∞ the capacitor behaves like at open circuit (R=∞).
Example:
100 V
100 V
12.5 A
100 V
2.5 A
10 A
0 V
Circuit
at t=0
at t=∞
Calculate initial currents. IB = 100 V/8 W = 12.5 A
Calculate final potentials.
* From Lecture 16
10/9/2018
Physics 122B - Lecture 23

21. Inductors: Early and Late
Initially, when a switch closes an inductor appears to have an infinite resistance and has a maximum potential drop across it. Ultimately, the inductor reaches a steady current flow with no potential drop across it. Therefore, at t=0 the inductor behaves like at open circuit (R=∞), and at t=∞ the inductor behaves like a short circuit (R=0). This behavior is opposite that of a conductor.
Example:
Circuit
at t=0
at t=∞
Calculate initial potentials.
Calculate final currents.
10/9/2018
Physics 122B - Lecture 23

22. Question Rank the order of the time constants in these circuits.
t1=t2=t3 ; (b) t1<t2<t3 ; (c) t2<t1<t3 ;
(d) t1>t2>t3 ; (e) t2>t1>t3 ;
10/9/2018
Physics 122B - Lecture 23

23. Lecture 23 Announcements
Lecture HW has been posted and is due on Wednesday at 10 PM.
Midterm examination 3 is this coming Friday---questions from lecture, tutorial AND Lab.
10/9/2018
Physics 122B - Lecture 23