LECTURE 25 AND RL CIRCUITS

Instructor: Kazumi Tolich Lecture 25

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¨ Reading chapter 23.7 to 23.9. ¤ Inductance ¤ Energy stored in magnetic field ¤ RL circuits Quiz: 1

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¨ The current is flowing upward on the front side of the coil and is increasing as a function of time. Are there an induced emf and induced current in the coil? If so, what is the effect of the induced current? A. No. B. Yes. It acts to further increase the rate of change of the current. C. Yes. It acts to decrease the rate of change of current.

� Quiz: 25-1 answer

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¨ Yes. It acts to decrease the rate of change of current.

¨ The increasing current creates increasing magnetic field that points to the left inside the coil.

¨ Due to the change in magnetic flux through the coil, an emf is induced.

¨ According to Lez’s law, the induced emf acts to oppose the change of the magnetic flux through the coil.

� Self-inductance

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¨ Self-inductance, or Inductance, �, is the proportionality constant that tells us how much emf will ∆ be induced for a given rate of change in current: ℰ = � . ∆

∆Φ � = � ∆�

¨ An object designed to have a large inductance is called an and is represented by the following symbol in a circuit diagram:

¨ The inductance depends only on the geometry.

¨ The inductance of a long solenoid with � turns of cross sectional area � and length ℓ is � = = � ��ℓ, where � = . ℓ ℓ Example: 1

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¨ 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 magnetic field

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¨ It takes energy to establish a current in an inductor; this energy is stored in the inductor’s magnetic field.

¨ The energy stored in an inductor with an inductance � carrying a current � is given by

1 � = �� 2

¨ This energy is stored in the magnetic field. The energy density of magnetic field is

� � = 2� Example: 2

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¨ 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/Demo: 1

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¨ A circuit containing a and an inductor is called an RL circuit. ¤ Current in a RL circuit cannot change discontinuously.

¨ The current in the circuit grows and approaches the final value after the switch is closed at � = 0.

ℰ � = 1 − �⁄ �

¤ where � = is the of the circuit. Quiz: 2

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¨ The time constant for the RL circuit is of the order of a few seconds. Describe what happens to the light bulb when the switch S is closed. A. The light bulb comes on immediately and then goes off in a few seconds. B. The light bulb comes on gradually in a few seconds and stays on. C. The light bulb does not come on at all. D. The light bulb comes on in a few seconds and then goes off instantaneously. E. The light bulb comes on in a few seconds and then goes off in a few seconds.

S bulb

+ L V - Quiz: 25-2 answer

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¨ The light bulb comes on gradually in a few seconds and stays on.

¨ Immediately after the switch is closed, the inductor acts like an open circuit.

¨ A long time after the switch is closed, the inductor acts like a short circuit.

¨ The current ramps up from zero to the final steady value.

S bulb

+ L V - Quiz: 3

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¨ Consider the circuit with a resistor with a resistance �, a light bulb with a resistance �, an ideal battery with an emf ℰ, an inductor with an inductance �, and a switch connected as shown. Initially the switch has been open for a long time. What is the current flowing in the inductor immediately after the switch is closed? ℰ A. ℰ B. ℰ C. D. 0 ℰ E. ℰ F. Quiz: 25-3 answer

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¨ 0

¨ Current in cannot change abruptly. Before the switch is closed, there is no current flowing in the inductor. Quiz: 4

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¨ Consider the circuit with a resistor with a resistance �, a light bulb with a resistance �, an ideal battery with an emf ℰ, an inductor with an inductance �, and a switch connected as shown. Initially the switch has been open for a long time. What is the current flowing in the light bulb immediately after the switch is closed? ℰ A. ℰ B. ℰ C. D. 0 ℰ E. ℰ F. Quiz: 25-4 answer

15 ℰ ¨ ¨ The current flows downward.

¨ The inductor acts like an open circuit immediately after the switch is closed.

¨ So, the resistor, the light bulb, and the battery can be treated as being connected in series immediately after the switch is closed.

¨ The equivalent resistance is � + �. Quiz: 5

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¨ Consider the circuit with a resistor with a resistance �, a light bulb with a resistance �, an ideal battery with an emf ℰ, an inductor with an inductance �, and a switch connected as shown. The switch has been closed for a long time. What is the current flowing in the light bulb? ℰ A. ℰ B. ℰ C. D. 0 ℰ E. ℰ F. Quiz: 25-5 answer

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¨ 0

¨ A long time after the switch is closed, the inductor acts like a short circuit.

¨ So, no current flows through the light bulb as the bulb can be treated as being connected in parallel with a wire.

¨ When the switch is closed, the light bulb turns on, and gradually dims and turns off eventually. Quiz: 6

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¨ Consider the circuit with a resistor with a resistance �, a light bulb with a resistance �, an ideal battery with an emf ℰ, an inductor with an inductance �, and a switch connected as shown. The switch has been closed for a long time. What is the current flowing in the inductor? ℰ A. ℰ B. ℰ C. D. 0 ℰ E. ℰ F. Quiz: 25-6 answer

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ℰ ¨ ¨ The current flows downward.

¨ A long time after the switch is closed, the inductor acts like a short circuit, so the inductor acts like a wire.

¨ So the circuit can be replaced with the battery connected to the resistor. Quiz: 7

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¨ Consider the circuit with a resistor with a resistance �, a light bulb with a resistance �, an ideal battery with an emf ℰ, an inductor with an inductance �, and a switch connected as shown. Initially the switch has been closed for a long time. What is the current flowing in the light bulb immediately after the switch is opened? ℰ A. ℰ B. ℰ C. D. 0 ℰ E. ℰ F. Quiz: 25-7 answer

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ℰ ¨ ¨ The current flows upward.

¨ The current flowing through an inductor cannot change abruptly. ℰ ¨ Just before the switch is opened, the current flowing in the inductor was downward. ¨ That current has to flow up the light bulb because there is no other path for the current to flow. Demo: 2

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¨ 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 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.