Wednesday, 10 August 2011

Physics: Faraday's Law and Alternating Current

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Electromagnetic Induction
Lenz’s Law states that an induced current is always in such a direction as to oppose the motion or change causing it. We say that in a non-uniform (or margin of) B-field, a non-zero relative motion between the circuit and magnetic field, the magnetic flux passing through the circuit changes, then an induced current along the coil is induced opposes the change in magnetic flux.
-          Magnetic flux of a coil in a uniform B-field is given by Φ = NBAcosθ, where N is the number of turns in coil, A is the area of coil and θ is the angle between (normal vector) of plane of coil and the direction of B-field. The unit is Weber (Wb).
-          By Faraday’s Law, the induced e.m.f. of circuit ε is equal to –dΦ/dt, where the negative sign indicates that it’s induced opposing the change (Lenz’s Law).
Example 1: When a metal ring drops vertically to a bar magnet (S upward), it induces anti-clockwise current to produce out-of-paper B-field as to oppose the increasing into-the-paper B-field. It has no induced current at middle since there’s no flux change; and clockwise current is induced when it leaves the magnet.
Example 2: When a rectangular coil (1 turn; x-dimension lx, y-dimension: ly) is pulled out of a uniform into-the-paper B-field of strength B at y-direction of speed v, then the direction of induced current is clockwise as to produce more into-the-paper B-field, the magnitude is dΦ/dt = d(BA)/dt = Blxd(ly)/dt = Blxv.
Note: we can use right-hand rule to solve the direction of induced current since left and right hand are mirror-image each other while F = -B x ε in electromagnetic induction.
Search coil is a small instrument with a coil of many turns to measure varying B-field. Before any measurement we should rotate the coil such that A//B, i.e., θ = 0. It’s quite sensitive as we can show the result on CRO and the area of search coil is very small (the magnitude of cm).
1)       Varying, usually sinusoidal magnetic field: assume B’ = B0sin ωt where B0 is the amplitude or peak of the magnetic field, then ε = -dΦ/dt = -NAdB’/dt = -NAB0ωcos ωt. By observing wave on CRO we can find its variation (B0 and frequency)
2)       Steady field: we put the search coil at the margin of the field or rotate the coil within the B-field and calculate the field density by ε =-NBdA/dt.
Applications
1)       Microphone: a diaphragm is connected to a coil with magnet inside it. When sound is emitted, the sound waves vibrate the diaphragm and cause the flux change within the coil, an induced current is generated and as a signal to be stored or playback elsewhere.
2)       Magnetic tape playback works in similar principle; when a sound is produced, the induced current connected to another iron core called the pick-up head magnetizes the type in a specified pattern; during playback, the magnetized pattern cause varying B-field at the iron core and produce the same pattern of varying current to produce the sound.
3)       Generator has similar set-up with a motor, but this time external force rotates the coil to generate current. A commutator is used such that d.c. is generated, but with a slip ring, a sinusoidal a.c. is produced.
 The induced current can be increased by
-          Stronger magnets (stronger B-field)
-          Using soft iron core inside the coil
-          Increasing number of turns
-          Rotating the coil faster
Note that energy conserves that when the induced current increases, the required power increases as well. Also in practical, magnet is rotated instead of coil because the friction between coil and commutator or slip rings causes sparks which is dangerous.
Eddy currents
In a bulk piece of metal entering/exiting B-field, an eddy current with closed loop of current within the metal is induced to oppose the motion (flux change) that produced them.
For example when a piece of metal goes out a into-the-paper B-field, (many) clockwise loop of current is induced inside the metal.

Application
-          Breaking effects: it slows down objects when entering/exiting the B-field. For example in pointer instruments, the coil is attached with a large metal piece so that a large eddy current is produced during it’s movements so that it stops at final reading steadily.
-          Induction heating: By using high frequency a.c., the fast change in magnetic flux cause a large eddy current on metal pot and hence heating it. It has no heating effect on non-metal at all, so it’s safer.
Unwanted eddy currents causes heat lost (energy lost) like the eddy currents in soft iron core in generator, so some eddy currents are minimized like laminated soft iron core is used in transformer that the path of eddy current for one completed loop is longer, hence higher resistance and smaller current, by P = VI, power loss is reduced.
Alternating current: the voltage that polarity of changes time to time.
There’re many types of a.c. V-t waveform, a typical one is called sinusoidal waveform which can be expressed by V = V0cos ωt, where V0 is the peak voltage (amplitude) and ω is the angular frequency (recall that ωT = 2π).
In a circuit with fixed resistance, V and I are in phase since V = IR, I = (V0/R)cos ωt = I0cos ωt.
Now denote <f(x)> as the average value of f.
Consider the average power of a alternating circuit of in sinusoidal waveform (if it’s not sinusoidal, the following calculation does not valid): <P> = <I2R> = <I2>R (since R is fixed)
Integrate I2 from 0 to 2π and divide by 2π(to find the average), we get <I2> = I02/2 and Irms = (<I2>)0.5 = I0/20.5 where Irms is the root-mean-square value of current in which a steady d.c. which Irms has the same power with an a.c. with current I.
Similarly, <V2> = V02/2, Vrms = V0/20.5, and <P> = VrmsIrms.
Note that for a.c. ammeter or supplies, they tend to show r.m.s. values instead of peak values.
Transformer
When one solenoid is connected to d.c. supply while another solenoid beside is connected to an a.c. ammeter, and the d.c. supply is suddenly turn on, there’s a flux change by the solenoid and hence induced current is detected. If a.c. is used, then the flux change spontaneously which makes the induced current continuous. At the same time, magnetic flux produced by the secondary coil may cause the cause in magnetic flux in primary coil and changes its current. This is called mutual induction.
In a transformer, two coils are connected to a soft iron core. In ideal case, all magnetic flux flows around the core with no leakage.

Considering dΦp/dt = dΦs/dt, -εp/Np = -εs/Ns, then Ns/Np = -εsp. When the resistance of coil is zero (ideal case), then V = ε that Vs/Vp = Ns/Np. We say voltage ratio = turns ratio.
Consider efficiency of transformer = VsIs/VpIp * 100%, if it’s ideal that VsIs = VpIp, then we have Vs/Vp = Ns/Np = Is/Ip as well. Typical practical efficiency is about 90%.
Loss of energy in transformer
-          Resistance of coil consumes energy
-          Eddy current in the soft iron coil, hence laminated soft iron coil is used.
-          Spontaneous magnetization of soft iron coil in opposite direction raises molecules’ k.e.
When secondary coil has more turns we call that a step-up transformer while if secondary coil has less turns than the primary coil, we call that a step-down transformer.
In practical, in power transmission, farther devices receives less energy since energy is consumed by the wire (it’s not ideal). By P = VI, when the voltage stepped-up, the current stepped-down, then it reduces heat loss on the wire.
In Hong Kong, electricity generated stepped up to several hundred kV, and then stepped-down to 132kV, 11kV in urban area, finally 220V at household

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