lecture10 - 10. Lecture 10 10.1 Force between currents We...

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Unformatted text preview: 10. Lecture 10 10.1 Force between currents We saw that a current circulating along a wire creates a magnetic field. If another current is in its vicinity it experiences a force. In fig.59 we see a simple configuration of two parallel wires carrying currents in the same direction. As discussed in the previous lecture, the magnetic field produced by wire 1 has the direction indicated in the figure and its intensity is µ0 ￿ |B | = I1 (10.1) 2π r According to the right-hand rule, the force experienced by the second wire is directed toward the first one (attractive force) and is µ0 L ￿ ￿ | F | = I 2 L| B | = I1 I2 2π r (10.2) where r is the distance between the cables, L their length and I1,2 the respective currents. Suppose they are separated by 1cm, they are 1m long and I1 = I2 = 10A. µ0 Using that 2π = 2 × 10−7 mT we find A ￿ |F | = 2 × 10−7 mT 1m 2 2 10 A = 2 × 10−3 mT A = 2 × 10−3 N A 1cm (10.3) A very small but measurable force. 10.2 Magnetic induction If we move a cable through a magnetic field, the charges in the cable will feel a force and then, if the cable forms a closed loop, a current will circulate. Consider the example of fig.60. The force on a charge inside the cable is ￿ |F | = qvB (10.4) ￿ Since the same force would be produced by an electric field of magnitude |E | = vB such force can move charges against a potential difference ∆V = vBL where L is the length of the cable. Namely if we have a resistor R in the rest of the circuit a current I = ∆V will circulate. This is in fact an electric generator!. Not very practical but R conceptually simple. In reality, it is better to move the cable in a circular motion so that it comes back to the original position and can keep moving. Notice that from the direction of the induced current we see that the moving cable acts as an effective battery whose positive terminal is in the lower cable. – 69 – Figure 59: Two parallel wires attract if current circulate in the same direction. They repel is the current circulates in opposite direction. A deeper insight into the problem can be obtained if we compute the magnetic flux through the loop. Since the magnetic field is constant and perpendicular to the loop, the flux is simply: ￿ Flux = Φ = |B |Ld (10.5) where L is the length of the moving conductor and d is its distance to the resistor. The distance d changes in time such that ∆d =v ∆t (10.6) where v is the velocity of the moving wire. Since the loop becomes bigger, namely d is growing we see that the flux changes as ∆Φ ￿ = |B |Lv ∆t (10.7) which is nothing else but the emf or voltage between the end points of the moving cable. Moreover, we see that the current generated in the cable will itself create a magnetic field in the loop which is opposite to the one already present, namely it creates an extra field that opposes the increase in flux. This phenomenon can be exemplified by a demo (see fig.61 and fig.62) where a loop of wire is moved in the proximity of a magnet and a current is detected by an Ammeter. The same experiment shows that if we move the – 70 – magnet instead of the loop of wire then the effect is the same. Although this appears more or less evident it is not clear in the latter case which force is moving the charges in the loop. Thinking about this problem was one of the reasons Einstein came up with the theory of relativity that we are going to discuss later in the course. Figure 60: When a conductor moves in a magnetic field, the charges inside it feel a force that induces a current. The resulting emf can be computed from the Lorentz force or from the Faraday law with identical results. Notice that the induced current also creates a magnetic field, indicated with the green arrows, which opposes the change in flux. For the moment we can summarize these finding in two important laws, the Faraday and Lenz law. Faraday’s law says that if the magnetic flux through a loop changes in time then an electromotive force is generated proportional to the rate of variation of the flux: ￿ ￿ ￿∆ Φ￿ ￿ ￿ |emf| = ￿ (10.8) ∆t ￿ where the Greek letter Φ is used to denote flux. It is completely equivalent as if we put a battery with ∆V = emf. It is important to know also in which direction the force goes or equivalently, in such effective battery which is the positive and which the negative pole. This is given by Lenz law. It says that the current generated in the loop creates – 71 – a magnetic field that opposes the change in flux. Namely if the flux increases, the magnetic field created by the current is opposite to the magnetic field already present and vice versa, if the flux decreases, the current tend to reinforce the magnetic field. This phenomenon is known as magnetic induction, a magnetic field induces a current. To exemplify this issues further we can use two interesting demos (see fig.63 and 64). In the first the current induced in a ring creates a repulsion that shoots the ring into the air. In the second the same repulsion is used to stop a pendulum showing how magnetic braking works. Figure 61: Demo. A current is induced in a loop if we move it in the presence of a magnetic field. See also fig. 62. – 72 – Figure 62: A current is induced in the loop if we either move the magnet or the loop. In both cases the magnetic flux through the loop changes. – 73 – Figure 63: Demo. Starting up the current through the coil induces a current on a ring producing a repulsive force and shooting the ring in the air. Cooling the ring decreases the resistance therefore increasing the induced current and the height reached by the ring. – 74 – Figure 64: Demo. The current induced in the pendulum when it goes through the poles of the magnet creates a repulsive force that brakes the pendulum halting it. The same principle can be used in electric car brakes. In that case, the induced current can charge the battery and the energy reused to propel the car. – 75 – ...
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This note was uploaded on 12/07/2011 for the course PHY 219 taught by Professor Na during the Fall '11 term at Purdue University.

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