LECTURE 19 INDUCTANCE & RL CIRCUITS Lecture 19 2 ¨ Reading chapter 28-6 to 28-8. ¤ Inductance ¤ Magnetic energy ¤ RL circuits Demo: 1 3 ¨ Two coils ¤ Demonstration of mutual induction. ¤ Intensity of induced current changes with separation. Mutual inductance 4 ¨ Current �" produces a magnetic flux through circuit 2 given by ¨ Similarly the magnetic flux through circuit 1 due to current �# is given by S1 ¨ We call M the mutual inductance of the two circuits. ¨ The mutual inductance depends on the geometrical arrangement of the two circuits. � for two coaxial solenoids ¨ The inner solenoid has a current �" flowing through �" turns in length �. The magnetic ) + - field is given by � = * , ,. " . ¨ This causes a magnetic flux through the outer # solenoid of �0"# = �#�" ��" = ��", ) 345+ + where � = * , , 5. Self-inductance 6 ¨ A current in a loop produces a magnetic flux through the loop proportional to the current. The constant of proportionality L is called the self-inductance in Henry, H. ¨ The self-inductance depends only on the geometry. ¨ Every loop has an inductance, but an object designed to have a large inductance is called an inductor and is represented by the following symbol in a circuit diagram: ¨ The inductance of a long solenoid with N turns of cross sectional area A and length ℓ is: Example: 1 7 ¨ An inductor is made by tightly winding a wire with a diameter d = 0.30 mm around a cylinder with a radius r = 2.0 mm. How long does the cylinder have to be to have an inductance of L = 10 µH? Quiz: 1 8 Self-induced emf 9 ¨ The potential difference across an inductor is given by Quiz: 2 Example: 2 11 ¨ A current I = 1.0 A passes through an inductor coil with L = 10.0 mH. What potential difference is induced across the coil if the current drops steadily to zero in Δt = 5.0 μs? Energy in inductors and magnetic fields 12 ¨ When magnetic field is created due to a current through an inductor, energy is stored in the magnetic field. ¨ An inductor with an inductance, L, carrying current, I, stores magnetic energy, Um, given by ¨ Magnetic energy density, um, is the magnetic energy per unit volume. Example: 3 13 ¨ A solenoid with N = 2000 turns has a cross-sectional area equal to A = 4.0 × 10-4 m2 and a length equal to l = 0.30 m. The solenoid carries a current of I = 4.0 A. a) Calculate the magnetic energy stored in the solenoid. b) Find the magnetic energy density in the solenoid. RL circuit 14 ¨ A circuit containing a resistor and an inductor is called an RL circuit. ¤ Current in a RL circuit cannot change discontinuously. Energizing an RL circuit 15 ¨ The switch is closed at t = 0. dI E0 − IR − L = 0 From Kirchhoff’s loop rule dt ¨ The current in the circuit grows and approaches the final value If. E I t 0 1 e−t /( L/ R) I 1 e−t /τ ( ) = ( − ) = f ( − ) R ¨ The time constant of the circuit is given by L τ = R Current after a short time or a long time 16 ¨ The switch was open, then closed. ¨ Immediately afterwards, the inductor can be treated as an open circuit. ¨ A long time after the switch is closed, the inductor can be treated as a short circuit. Quiz: 3 De-energizing an RL circuit 18 ¨ The switch has been at position e for a long time, then switched to position f at t = 0. ¨ The current in the inductor starts out at I0 and exponentially decays. dI −IR − L = 0 From Kirchhoff’s loop rule dt E I t I e−t/τ 0 e−t/τ ( ) = 0 = R1 + R Demo: 2 19 ¨ RL circuit ¤ An inductor in series with a resistor, a battery and a switch. ¤ When the switch is closed the current rises slowly from zero to a steady- state value. E I t 0 1 e−t /( L/ R) I 1 e−t /τ ( ) = ( − ) = f ( − ) R E I t I e−t/τ 0 e−t/τ ( ) = 0 = R1 + R Example: 4 20 ¨ 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 μs? b) At what time has the current decayed to 1% of its original value? Demo: 3 21 ¨ Lamp in parallel with solenoid ¤ A large DC current introduced suddenly to a large inductor cannot pass through the coil at first, so a lamp in parallel with the coil lights up. ¤ After the current becomes steady, the coil draws more current and bulb dims. ¤ When the current is switched off suddenly, the induced voltage in the coil (back EMF) again lights the lamp. ¤ A separate neon lamp in parallel with the coil shows that the direction of the second voltage surge is the opposite of the first. Make-before-break switch 22 ¨ A make-before-break switch allows us to remove the battery from the circuit without sparks. ¨ If the contact e was broken before making a contact with f, the current in the inductor has nowhere to go..