ch22_csnk

Chapter 22 - Induction and Alternating Current __Important Equations to Know__ math emf = -N\frac{\Delta[ABcos(\theta)]}{\Delta t} math math \frac{V_2}{V_1} = \frac{N_2}{N_1} math math \frac{V_m_a_x}{\sqrt{2}} = V_r_m_s math math \frac{I_m_a_x}{\sqrt{2}} = I_r_m_s math __Terms and Laws__ electromagnetic induction emf Lenz's Law Faraday's Law generator rms vs. maximum (current, potential difference) alternating and direct current motors back emf mutual inductance transformers __Main Points__
 * Changing the magnetic field strength near a conductor induces an emf.
 * The direction of an induced current in a circuit is such that its magnetic field opposes the change in the applied magnetic field.
 * Generators use induction to convert mechanical energy into electrical energy.
 * Alternating current is measured in terms of rms current.
 * Motors use an arrangement similar to that of generators to convert electrical energy into mechanical energy.
 * Mutual inductance involves the induction of a current in one circuit by means of changing current in a nearby circuit.
 * Transformers change the potential difference of an alternating current.

__Electromagnetic Induction__  You don't need wires or a power supply to create a current. Instead, you can move a wire in a magnetic field to produce a current. This process is referred to as **electromagnetic induction. Electromagnetic induction** is defined as producing an emf in a circuit by changing the magnetic field. To change the field, you can change the position, strength, or orientation of the external magnetic field. To review, **emf** is rate of change of magnetic flux on the loop. This is very similar to potential difference. Basically, electromagnetic induction works like this: 1. Magnetic fields make moving charges deflect. 2. Charges moving with a velocity at an angle to a magnetic field experience a force (right hand rule). 3. This force is a lot like potential difference, because it makes the charges move. 4. Remember, the wire must be moving and it must cross magnetic field lines.

__Lenz's Law__ Emf is greatest when the plane of the loop is perpendicular to magnetic field lines. It decreases and becomes zero as the loop rotates to become parallel to the field lines. Also, an increase in the number of field lines (which is increasing the field strength) or increase in the area of the loop will increase the induced current.

The current induced will be opposed by the magnetic field it approaches. As the current's magnetic field get stronger, so does the opposition to it, which leads to Lenz's law: //<span style="color: #000000; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">The magnetic field of the induced current opposes the change in the applied magnetic field. // <span style="color: #000000; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">When the magnet moves away, the field lessens, as does the opposition to it. We can use this law to find the direction of an induced current. To find the emf, we use **Faraday's Law of Induction.**

__Faraday's Law__ <span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> math emf = -N\frac{\Delta[ABcos(\theta)]}{\Delta t} math N is the number of turns in a coil; a negative sign preceds it to indicate the polarity of the induced emf (Lenz's law). A is the area of the coil. B is the field strength. Theta is the angle of orientation, or the angle between the normal of the loop and the field lines. Delta t is the change in time. <span style="display: block; font-size: 240%; color: #dd3c3c; font-family: 'Trebuchet MS', Helvetica, sans-serif; text-align: center;">PRACTICE <span style="color: #000000; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> 3. A circular wire loop with a radius of 0.33 m is located in an external magnetic field of strength 0.25 T that is perpendicular to the plane of the loop. The field strength changes to -0.35 T in 1.5s. Find the magnitude of the average induced emf during the interval (Faughn 800). math r = 0.33 m math math B_i = 0.25T math math B_f = -0.35T math math \Delta t = 1.5s math math N = 1 math

math A = \pi r^2 math math A = \pi (0.33m)^2 math math A = 0.1089 \pi m^2 math

math emf = -N\frac{\Delta[ABcos(\theta)]}{\Delta t} math math emf = -N\frac{\Delta[AB]}{\Delta t} math math emf = -1\frac{[0.1089 \pi m^2 (0.35T + 0.25T)}{1.5s} math math emf = - 0.14 V math

<span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> __Applications of Induction__ Induction has a variety of applications, including doorbells and tape recorders. In a doorbell, when the button is pressed, the light bulb behind the button goes out, signaling a break in the circuit. This circuit break is used to induce a series current on coils of wire, which force an iron plunger in a chime. In a tape recorder, sound waves are converted into electric current. The tape recorder has an iron ring with a gap with wires wrapped around it. As the current passes through the wire, the changes in the magnetic field are recorded on the magnetic tape of the recorder. They can then be played back and converted into sound waves.

<span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> __Generator__ <span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">Another application of induction is the **generator.** Moving a wire throughout a magnetic field induces a current. Twirling the wire will have the same effect as moving the wire in and out of the field. Moving the wire is an example of mechanical motion. Instead of having a person generate this motion, one can connect the wire to a device that will generate the motion.

When the loop of wire is perpendicular to the magnetic field, it is parallel to the field and does not cross field lines. This means an emf will not be generated. However, when the loop is parallel to the field, its wires will cross the magnetic field lines, and an emf will be generated.

<span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">(Faughn 804). As the loop rotates, the emf it creates changes direction. The emf moves from zero to positive to zero to negative. In part a, the two main sections of the wire are parallel to the field lines, so no emf is induced. B and d are outside of the field, so no emf can be induced upon them. As the wire continuously moves, the emf is continuously changing.

__Alternating Current__ <span style="font-size: 10pt; color: black; font-family: 'Lucida Sans Unicode'; msofareastfontfamily: 'Times New Roman'; msoansilanguage: EN-US; msofareastlanguage: EN-US; msobidilanguage: AR-SA;">Because the emf changes constantly, it produces a constantly changing current. This current is called ** alternating current ** because it changes direction every 1/60th of a second. This change is not noticeable, because it happens too fast for the human eye to perceive. Note: Resistors work in both alternating and direct current because they resist the flow of current, regardless of its direction.

__Rms and Max (Current and Potential Difference)__ Current and potential difference fluctuate from a maximum to zero to a minimum. To get a consistent measurement, we use an average called **rms**, or **root mean square**. **Rms current** is the amount of direct current that would dissipate the same amount of energy as alternating current in a resistor at the same cycle. Current is measured in rms values, so the maximum value is actually larger than the rms value. The relationship between rms current and maximum current is: math \frac{I_m_a_x}{\sqrt{2}} = I_r_m_s math math \frac{V_m_a_x}{\sqrt{2}} = V_r_m_s math In the US, potential difference is measured in rms values at 120 Volts. The maximum value possible, however is about 170 Volts. Note: Ohm's law holds for rms values.
 * Rms potential difference** has similar relationship to maximum potential difference as rms current has to maximum current.

<span style="display: block; font-size: 240%; color: #dd3c3c; font-family: 'Trebuchet MS', Helvetica, sans-serif; text-align: center;">PRACTICE <span style="display: block; font-size: 240%; color: #dd3c3c; font-family: 'Trebuchet MS', Helvetica, sans-serif; text-align: center;"> <span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">1. What is the rms current in a light bulb that has a resistance of 25 Ω and a rms potential difference of 120 V ? What are the maximum values for current and potential difference? (Faughn 810). math r = 25 \Ohm math math \Delta V_r_m_s = 120 V math

math \Delta V_r_m_s = \frac{\Delta V_m_a_x}{\sqrt{2}} math math \sqrt{2} \Delta V_r_m_s = \Delta V_m_a_x math math \sqrt{2} * 120 V = \Delta V_m_a_x math math 170 V = \Delta V_m_a_x math

math I_r_m_s = \frac{\Delta V_r_m_s}{R} math math I_r_m_s = \frac{120V}{25 \Omega} math math I_r_m_s = 4.8 A math math I_r_m_s = \frac{I_m_a_x}{\sqrt{2}} math math I_m_a_x = I_r_m_s * \sqrt{2} math math I_m_a_x = 4.8 A * \sqrt{2} math math I_m_a_x = 6.8 A math <span style="color: #000000; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> __<span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">Motors __ <span style="color: #13aaaa; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> __Mutual Inductance__ <span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">Two coils are wrapped around an iron ring. When the magnetic field in the primary changes, it induces an emf in the secondary. This is **mutual inductance**, or that ability of one current carrying circuit to influence an emf in a nearby circuit. The relationship between emf, M (constant for mutual inductance based on the system), current, and time using the following equation. <span style="color: #000000; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;"> math emf = -M\frac{\Delta I}{\Delta t} math
 * Motors** work in the opposite direction of generators, they convert electrical energy into mechanical energy. It is the same set-up as an alternating current generator. As the coil in the motor rotates, it induces a **back emf** that acts like friction and reduces the current powering the motor.

__Transformers__ Transformers use the principle of mutual inductance to convert potential differences of alternating currents to different strengths. They consist of a primary and secondary with N turns and V potential difference respectively.

math \frac{V_2}{V_1} = \frac{N_2}{N_1} math

If the secondary has more turns than the primary, it is a step-up transformer because it increases the potential difference. If the primary has more turns, the transformer is a step-down transformer because it reduces the potential difference. Transformers **DO NOT** violate the conservation of energy laws because the input power and output power are ideally the same.

In the real world, however, some power is lost due to small currents induced by changing magnetic fields in the iron core of the transformer, and due to resistance. Input power will always then be equal or greater than output power.

<span style="display: block; font-size: 240%; color: #dd3c3c; font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif; text-align: center;">PRACTICE 5. A television picture tube requires a high potential difference, which in older models is provided by a step-up transformer. The transformer has 12 turns in its primary and 2550 turns in its secondary. If 120 V is placed across the primary, what is the output potential difference? (Faughn 818).

math N_1 = 12 turns math math N_2 = 2550 turns math math \Delta V_1 = 120 V math math \frac{V_2}{V_1} = \frac{N_2}{N_1} math math \Delta V_2 = \frac{\Delta V_1 N_2}{N_1} math math \Delta V_2 = \frac{120V * 2550 turns}{12 turns} math math \Delta V_2 = 2.55 * 10^4 V math <span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">


 * INDEPENDENT PRACTICE**

<span style="font-family: 'Lucida Sans Unicode', 'Lucida Grande', sans-serif;">1. An AC generator has a maximum output emf of 7.25 * 10 3  V. What is the rms potential difference for this generator? (5.13 * 10 3 V)

2. A step up transformer used on a 240 V line has 27 turns on the primary and 8542 turns on the secondary. What is the potential difference across the secondary? (7.6 * 10 4 V)

3. A loop of wire with 25 turns and an area of 0.0243 square meters is perpendicular to a magnetic field of strength 2.85 T. If the loop is removed from the field in a time interval of 1.20 seconds, what is the average emf in the wire? (-1.44 V)

Self Induction, effective current, and maximum emf for a generator were not covered in class or tested.


 * CITATIONS**

Faughn, Jerry S. and Raymond A. Serway. __Physics__. New York: Holt, 2004. Strong, Tom. Course notes. Honors Physics, Dept. of Science, Mount Lebanon High School. May and June 2009.