1. Chapter 32 Inductance L and the stored magnetic energy RL and LC circuits RLC circuit

2. Resistance, Capacitance and Inductance Ohm’s Law defines resistance: Capacitance, the ability to hold charge: Capacitors store electric energy once charged: Resistors do not store energy, instead they transform electrical energy into thermo energy at a rate of: Inductance, the ability to “hold” current (moving charge). Inductors store magnetic energy once “charged” with current, i.e., current flows through it.

3. Inductance, the definition I When a current flows through a coil, there is magnetic field established. If we take the solenoid assumption for the coil: E ELEL + – When this magnetic field flux changes, it induces an emf, E L, called self-induction: or: This defines the inductance L, which is constant related only to the coil. The self- induced emf is generated by current flowing though a coil. According to Lenz Law, the emf generated inside this coil is always opposing the change of the current which is delivered by the original emf. For a solenoid: Where n: # of turns per unit length. N: # of turns in length l. A: cross section area V: Volume for length l.

4. Inductor We used a coil and the solenoid assumption to introduce the inductance. But the definition holds for all types of inductance, including a straight wire. Any conductor has capacitance and inductance. But as in the capacitor case, an inductor is a device made to have a sizable inductance. An inductor is made of a coil. The symbol is Once the coil is made, its inductance L is defined. The self-induced emf over this inductor under a changing current I is given by:

5. Unit for Inductance The SI unit for inductance is the henry (H) Named for Joseph Henry: 1797 – 1878 American physicist First director of the Smithsonian Improved design of electromagnet Constructed one of the first motors Discovered self-inductance

6. Discussion about Some Terminology Use emf and current when they are caused by batteries or other sources Use induced emf and induced current when they are caused by changing magnetic fields When dealing with problems in electromagnetism, it is important to distinguish between the two situations

7. Example: Inductance of a coaxial cable Start from the definition We have So the inductance is

8. Put inductor L to use: the RL Circuit An RL circuit contains a resistor R and an inductor L. There are two cases as in the RC circuit: charging and discharging. The difference is that here one charges with current, not charge. Charging: When S 2 is connected to position a and when switch S 1 is closed (at time t = 0), the current begins to increase Discharging: When S 2 is connected to position b. PLAY ACTIVE FIGURE

9. RL Circuit, charging Applying Kirchhoff’s loop rule to the circuit in the clockwise direction gives Here because the current is increasing, the induced emf has a direction that should oppose this increase. Solve for the current I, with initial condition that I( t=0 ) = 0, we find Where the time constant is defined as:

10. RL Circuit, discharging When switch S 2 is moved to position b, the original current disappears. The self-induced emf will try to prevent that change, and this determines the emf direction (Lenz Law). Applying Kirchhoff’s loop rule to the previous circuit in the clockwise direction gives Solve for the current I, with initial condition that we find

11. Energy stored in an inductor In the charging case, the current I from the battery supplies power not only to the resistor, but also to the inductor. From Kirchhoff’s loop rule, we have Multiply both sides with I: This equation reads: power battery =power R +power L So we have the energy increase in the inductor as: Solve for U L :

12. Stored energy type and the Energy Density of a Magnetic Field Given U L = ½ L I 2 and assume (for simplicity) a solenoid with L =  o n 2 V Since V is the volume of the solenoid, the magnetic energy density, u B is This applies to any region in which a magnetic field exists (not just the solenoid) So the energy stored in the solenoid volume V is magnetic (B) energy. And the energy density is proportional to B 2.

13. RL and RC circuits comparison RLRC Charging Discharging Energy Magnetic fieldElectric field Energy density

14. Energy Storage Summary Inductor and capacitor store energy through different mechanisms Charged capacitor Stores energy as electric potential energy When current flows through an inductor Stores energy as magnetic potential energy A resistor does not store energy Energy delivered is transformed into thermo energy

15. LC Circuits LC: circuit with an inductor and a capacitor. Initial condition: either the C or the L has energy stored in it. The “show” starts: when the switch S closes, t = 0 and the time starts. Your physics intuition: neither C nor L consumes energy, the initially stored energy will oscillate between the C and the L.

16. LC Circuits, the calculation Initial condition: Assume that the capacitor was initially charged to Q max. when the switch S closes, t = 0 and the time starts. Here q is the charge in the capacitor at time t. Because charges flow out of the capacitor to form the current I, we have: Apply Kirchhoff’s loop rule: Combine these two equations: Solve for the current I:with Here we also have

17. LC Circuits, the oscillation of charge and current Oscillations: simply plot the results, we find out that the charge stored in the capacitor and the current “stored” in the inductor oscillate. The phase difference is T/2. This means that when the capacitor is fully charged, the current is zero. When the capacitor has no charges in it, the current reaches its maximum in magnitude through the inductor. q From the formulas for the energies stored in a capacitor and an inductor, we know that this oscillation happens between electric energy and magnetic energy.

18. LC Circuits, the oscillation of energy From the following four formulas We have the oscillation of the energies in the capacitor and the inductor: From energy conservation:

19. Move from the ideal LC circuit to the real-life RLC circuit In actual circuits, there is always some resistance, therefore, there is some energy transformed to thermo energy by the resistance in the system and dissipates to the environment. Radiation is also inevitable in this type of circuit, and energy will be radiated out of the LC system as electromagnetic wave through space. The total energy in the circuit continuously decreases as a result of these processes Here we will only discuss about the energy dissipated through the resistance.

20. The RLC Circuit and the analysis Concentrating the resistance in the system into a resistor, with the inductor and the capacitor, we model the circuit with an RLC Circuit. PLAY ACTIVE FIGURE The capacitor is charged with the switch at position a. At time t = 0, the switch is thrown to position b to form the RLC circuit. Apply Kirchhoff’s loop rule: We have: Solve for q: And:

21. Mutual Inductance The magnetic flux through the area enclosed by a circuit often varies with time because of time-varying currents in nearby circuits This process is known as mutual induction because it depends on the interaction of two circuits

22. Mutual Inductance and transformers The current in coil 1 sets up a magnetic field that varies as I 1. When magnetic field lines pass through coil 2, cause the magnetic flux in coil 2 to change and induce current I 2 in coil 2. This process is called mutual inductance. If coil 1 has a current I 1 and N 1 turns, and coil 2 has N 2 turns. When the field lines that go through coil 1 completely go through coil 2, we have a transformer. Coil 1 and 2 are called prime and second coils. The terminal voltages at these two coils are If coil 2 connects to a resistor R 2, the resistance coil 1 “sees” is From energy conservation: We have: