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Chapter Review

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23-8   RL Circuits

A coil with a finite inductance (and negligible resistance) is called an inductor. Inductors are often used as circuit elements; a circuit with a resistor and an inductor in series is called an RL circuit. Recall that induction in a coil tends to resist changes in the current. As a result of this, the current in a circuit that contains an inductor does not rise and fall as quickly as it otherwise would; the inductance causes it to rise and fall gradually, the larger the inductance the more gradual the change in current.

Similar to what happens in RC circuits, when an emf is first applied (or shut off) in an RL circuit, the current increases (or decreases) exponentially with a characteristic time given by the time constant

.

For a current that is building up from zero in a RL circuit the result is

For a current that is falling off from the maximum value I_max = E/R the result is

We can see that in the case where the current is falling off, t represents the amount of time it takes for the current to fall to (e^-1)I_max = (0.368)I_max, that is, to 36.8% of its maximum value.

Physlet Illustration: RL Circuit
R =  W L =  H
In this simulation, a resistor and an inductor are connected in series to a 12-Volt battery. When the switch closes, the current in the circuit begins to flow. An ammeter is included in the circuit, and a graph of the current  vs. time is also shown. Vary the values of the resistor (1 W < R < 10 W) and inductor (1 H < L < 5 H). How do these affect the flow of current through the circuit?
Hints: As you increase or decrease R, what happens to the time it takes for the current in the circuit to approach its maximum value? As you increase or decrease L, what happens to the time it takes for the current in the circuit to approach its maximum value? What is the maximum current attained? Why? How does the current flowing through the inductor affect the circuit?

Practice Quiz

An RL circuit contains an inductor of inductance L and a resistor of resistance R giving a time constant of value t. If the resistance is doubled, the time constant will be... t/4 t/2 t 2t 4t

Which of the following statements is true concerning the current in an RL circuit? The time constant is how long it takes the current to build up to 36.8% of its maximum value. The time constant is how long it takes the current to build up to 63.2% of its maximum value. The time constant is how long it takes the current to fall off to 63.2% of its maximum value. The time constant is how long it takes the current to fall to 36.8% of its maximum value from 63.2% of it. The time constant is how long it takes the current to build up to 63.2% of its maximum value from 36.8% of it.

23-9   Energy Stored in a Magnetic Field

Because the induced current in an inductor resists the build up of current in the coil, it therefore requires energy to build up the current in an inductor against this "resistance." Once the current has been established, so has the magnetic field within the coil. The energy needed to build up this current is stored in the magnetic field. For an inductor of inductance L, sustaining a current I, the amount of energy U (the text uses E) stored in the magnetic field is

.

Just like with the energy stored in the electric field of a capacitor, the relationship between the energy stored in an inductor and its magnetic field is nicely expressed in terms of the energy density (energy per unit volume) of the magnetic field

.

This expression for the energy density applies to any magnetic field, not just those for an inductor.

Practice Quiz

A solenoid of inductance L sustains a current I storing an amount of energy U. If the current in the solenoid is reduced to half its previous value, the energy stored in the solenoid will be... U/4 U/2 U 2U 4U

23-10   Transformers

A very important practical device that relies on the phenomenon of induction is the transformer. A transformer is a device that uses induction to increase or decrease the voltage in a circuit. A transformer consists of a primary coil, containing N_p turns, across which an AC voltage V_p is applied. The primary coild is bound by an iron core to a secondary coil, containing N_s turns. By induction there will be an induced potential differece V_s across the secondary coil. The relationship between the primary and secondary voltages is given by the transformer equation:

.

When V_s > V_p we have a step-up transformer and when V_s << V_p we have a step-down transformer.

By conservation of energy, the average power in the secondary circuit must equal the average power in the primary circuit, I_sV_s = I_pV_p. Therefore,

.

So that a step-up transformer steps down the current and vice versa.

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Exercise 23.5   A Big Step Down:   Suppose you want to use one transformer to step down from a power line voltage of 50,000 V to a household voltage of 120 V. If your primary coil consists of 10,000 turns, how many turns would you need in your secondary coil?

Solution:   We are given the following information:

Given: V_p = 50,000 V, V_s = 120 V, N_p = 10,000     Find: N_s

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The transformer equation gives,

Insight   In practice the voltage from power lines is stepped down in several stages, not all at once.

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Practice Quiz

The primary coil of a transformer contains 100 turns and the secondary coil contains 200 turns. If the current in the primary coil is I, what it the current in the secondary coil? I 2I I/2 I/4 4I

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