23.1 Induced Emf and Magnetic Flux
- The crucial quantity in induction is magnetic flux , defined to be , where is the magnetic field strength over an area at an angle with the perpendicular to the area.
- Units of magnetic flux are .
- Any change in magnetic flux induces an emf—the process is defined to be electromagnetic induction.
23.2 Faraday’s Law of Induction: Lenz’s Law
- Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
when flux changes by in a time .
- If emf is induced in a coil, is its number of turns.
- The minus sign means that the emf creates a current and magnetic field that oppose the change in flux —this opposition is known as Lenz’s law.
23.3 Motional Emf
- An emf induced by motion relative to a magnetic field is called a motional emf and is given by where is the length of the object moving at speed relative to the field.
23.4 Eddy Currents and Magnetic Damping
- Current loops induced in moving conductors are called eddy currents.
- They can create significant drag, called magnetic damping.
23.5 Electric Generators
- An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by where is the area of an -turn coil rotated at a constant angular velocity in a uniform magnetic field .
- The peak emf of a generator is
23.6 Back Emf
- Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.
- Transformers use induction to transform voltages from one value to another.
- For a transformer, the voltages across the primary and secondary coils are related by where and are the voltages across primary and secondary coils having and turns.
- The currents and in the primary and secondary coils are related by .
- A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.
23.8 Electrical Safety: Systems and Devices
- Electrical safety systems and devices are employed to prevent thermal and shock hazards.
- Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
- The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
- A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
- An isolation transformer insulates the device being powered from the original source, also to prevent shock.
- Many of these devices use induction to perform their basic function.
- Inductance is the property of a device that tells how effectively it induces an emf in another device.
- Mutual inductance is the effect of two devices in inducing emfs in each other.
- A change in current in one induces an emf in the second: where is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
- Symmetrically, a change in current through the second device induces an emf in the first: where is the same mutual inductance as in the reverse process.
- Current changes in a device induce an emf in the device itself.
- Self-inductance is the effect of the device inducing emf in itself.
- The device is called an inductor, and the emf induced in it by a change in current through it is where is the self-inductance of the inductor, and is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
- The unit of self- and mutual inductance is the henry (H), where .
- The self-inductance of an inductor is proportional to how much flux changes with current. For an -turn inductor,
- The self-inductance of a solenoid is where is its number of turns in the solenoid, is its cross-sectional area, is its length, and is the permeability of free space.
- The energy stored in an inductor is
23.10 RL Circuits
- When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is where is the final current.
- The characteristic time constant is , where is the inductance and is the resistance.
- In the first time constant , the current rises from zero to , and 0.632 of the remainder in every subsequent time interval .
- When the inductor is shorted through a resistor, current decreases as Here is the initial current.
- Current falls to in the first time interval , and 0.368 of the remainder toward zero in each subsequent time .
23.11 Reactance, Inductive and Capacitive
- For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a phase angle.
- The opposition of an inductor to a change in current is expressed as a type of AC resistance.
- Ohm’s law for an inductor is where is the rms voltage across the inductor.
- is defined to be the inductive reactance, given by with the frequency of the AC voltage source in hertz.
- Inductive reactance has units of ohms and is greatest at high frequencies.
- For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a phase angle.
- Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is where is the rms voltage across the capacitor.
- is defined to be the capacitive reactance, given by
- has units of ohms and is greatest at low frequencies.
23.12 RLC Series AC Circuits
- The AC analogy to resistance is impedance , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
- Impedance has units of ohms and is given by .
- The resonant frequency , at which , is
- In an AC circuit, there is a phase angle between source voltage and the current , which can be found from
- for a purely resistive circuit or an RLC circuit at resonance.
- The average power delivered to an RLC circuit is affected by the phase angle and is given by is called the power factor, which ranges from 0 to 1.