Ch. 23 Electromagnetic Induction, AC Circuits, And Electrical Technologies


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1 Ch. 23 Electromagnetic Induction, AC Circuits, And Electrical Technologies Induced emf  Faraday s Experiment When a magnet moves toward a loop of wire, the ammeter shows the presence of a current When the magnet moves away from the loop, the ammeter shows a current in the opposite direction If polarity of the magnet is reversed, the direction of the current is reversed When the magnet is held stationary, there is no current Magnetic Flux Faraday s conclusion: an electrical current is produced by a changing magnetic field (as if a source of emf were connected to the secondary circuit for a short time) The emf is actually induced by a change in the quantity called the magnetic flux in the loop of wire, rather than simply by a change in the magnetic field B = B A = B A cos θ θ is the angle between B and the normal to the plane 1
2 Magnetic Flux When the field is perpendicular to the plane of the loop, as in a, θ = 0 and Φ B = Φ B, max = BA When the field is parallel to the plane of the loop, as in b, θ = 90 and Φ B = 0 The flux can be negative, for example if θ = 180 The flux can be visualized with respect to magnetic field lines: The value of the magnetic flux is proportional to the total number of lines passing through the loop SI units of flux are T m² = Wb (Weber) 0 surface B closed Faraday s Law and Electromagnetic Induction The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit If a circuit contains N tightly wound loops and the flux changes by ΔΦ during a time interval Δt, the average emf induced is given by Faraday s Law: N t B The negative sign in Faraday s Law is included to indicate the polarity of the induced emf, which is found by Lenz Law The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. 2
3 Application of Faraday s Law Motional emf The electrons in the conductor experience a magnetic force F = q v B As a result of this charge separation, an l electric field is produced in the conductor. Charges build up at the ends of the conductor until the downward magnetic force is balanced by the upward electric force qvb = qe E vb The potential difference between the ends of the conductor can be found by ΔV = E l = B l v Note: V is independent of the charge density of the conductor. Bv Motional emf in a Circuit (a) A motional emf = Blv is induced between the rails when this rod moves to the right in the uniform magnetic field. The magnetic field B is into the page, perpendicular to the moving rod and rails and, hence, to the area enclosed by them. (b) Lenz s law gives the directions of the induced field and current, and the polarity of the induced emf. Since the flux is increasing, the induced field is in the opposite direction, or out of the page. RHR2 gives the current direction shown, and the polarity of the rod will drive such a current. RHR1 also indicates the same polarity for the rod. (Note that the script E symbol used in the equivalent circuit at the bottom of part (b) represents emf.) and Bv I R 3
4 Lenz Law The polarity of the induced emf is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. Example 37. (a) If the emf of a coil rotating in a magnetic field is zero at t=0, and increases to its first peak at t=0.100 ms, what is the angular velocity of the coil? (b) At what time will its next maximum occur? (c) What is the period of the output? (d) When is the output first onefourth of its maximum? (e) When is it next onefourth of its maximum? 4
5 Magnetic Dragging and Damping As it enters and leaves the field, the change in flux produces an eddy current. Magnetic force on the current loop opposes the motion. There is no current and no magnetic drag when the plate is completely inside the uniform field. damping is greatest for large oscillations and goes to zero as the motion stops. Generators An electric generator converts mechanical energy into electric energy. An outside source of energy is used to turn the coil, thereby generating electricity. E NBA sint A split ring commutator generates DC power 5
6 Motors An electric motor is exactly the opposite of a generator it uses the torque on a current loop to create mechanical energy. Transformers A transformer is used to change voltage in an alternating current from one value to another. p: primary coil S: secondary The power in both circuits must be the same; therefore, if the voltage is lower, the current must be higher. 6
7 Inductance When the switch is closed in this circuit, a current is established that increases with time. Inductance is the proportionality constant that tells us how much emf will be induced for a given rate of change in current: Solving for L, L N I B N I B Direction of Induced Potential From the definition, the inductance of a solenoid can be calculated: 7
8 RL Circuit I IR L t 0 When the switch is closed, the current immediately starts to increase. The back emf in the inductor is large, as the current is changing rapidly. As time goes on, the current increases more slowly, and the potential difference across the inductor decreases. The current in an RL circuit varies exponentially as a function of time. I R 1 e t / The time constant is: Why Exponential Form Again? I IR L t 0 R I I to go I to go I I R to go I R L t e 1 e to go e R ( R / L) t ( R / L) t t / L R rate of change of I to go is proportional to the value of I to go. 8
9 Alternating Voltages and Currents A ground fault circuit interrupter can cut off the current in a short circuit within a millisecond. Capacitor in an AC Circuit Consider a circuit containing a capacitor and an AC source. The voltage across the plates follows the supplied AC voltage. The charge on the plates also follows (q=cv) The current flowing in the external circuit is what is required to adjust the charge on the capacitor. This current precedes the voltage. The voltage lags the current by 90 The impeding effect of a capacitor, the capacitive reactance, is given by X C 1 C is in Hz, C is in F, X C in ohms Ohm s Law for a capacitor in an AC circuit V rms = I rms X C 9
10 Inductors in AC Circuits The voltage across an inductor leads the current by 90. X L = L The power factor for an RL circuit is: Currents in resistors, capacitors, and inductors as a function of frequency: The RLC Series Circuit The current in the circuit varies sinusoidally with time The instantaneous voltage across the resistor is in phase with the current The instantaneous voltage across the inductor leads the current by 90 The instantaneous voltage across the capacitor lags the current by 90 10
11 RLC Circuit The phase angle for an RLC circuit is: If X L = X C, the phase angle is zero, and the voltage and current are in phase. The power factor: Summary of Circuit Elements, Impedance and Phase Angles 11
12 Generating An AC Signal With Circuitry We know that this leads to exponential time dependence. If we first charge a capacitor and use it as a battery, what is the time dependence of the current in the resultant circuit? In a real circuit, there is always some resistance,... Resonance in an AC Circuit Resonance occurs at the frequency, ƒ o, where the current has its maximum value Minimum impedance occurs when X L = X C o 1 LC ƒ o 1 2 LC Theoretically, if R = 0 the current would be infinite at resonance Real circuits always have some resistance Analogy with massspring SHM: C k 1 ; L m 12
13 Examples 74. (a) What is the characteristic time constant of a 25.0 mh inductor that has a resistance of 4.00? (b) If it is connected to a 12.0 V battery, what is the current after 12.5 ms? 101. An RLC series circuit has a 2.50 resistor, a 100 H inductor, and an 80.0 F capacitor. (a) Find the circuit s impedance at 120 Hz. (b) Find the circuit s impedance at 5.00 khz. (c) If the voltage source has a V rms =5.60 V, what is I rms at each frequency? (d) What is the resonant frequency of the circuit? (e) What is I rms at resonance? Chapter 23, Summary 1. Magnetic flux. 2. Faraday s law of induction. 3. Lenz s law and motional emf 4. Generators and motors 5. Inductance 6. Transformers 7. Impedance of LCR circuits 8. Phase angle and power factor 9. Resonant frequency. 13
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