lecture9 - 9. Lecture 9 9.1 Magnetic forces on an electric...

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Unformatted text preview: 9. Lecture 9 9.1 Magnetic forces on an electric current Since an electric current is charges in motion it follows that a wire through which a current circulates will experience a force in the presence of a magnetic field. In fact an interesting demo (fig.52) makes this effect evident. More precisely, suppose that, inside the conductor, an electric charge density (charge per unit volume) ρ is moving with velocity v . Then, during time ∆t all the charge contained in a volume v ∆tA will pass through a cross section of area A as can be seen in fig.53. The electric current, namely the amount of charge going through a section of the conductor of area A per unit time is therefore ρv ∆tA ∆Q I= = = ρvA (9.1) ∆t ∆t On the other hand the magnetic force on a moving charge is ￿ ￿ |F | = q |￿ ||B | sin θ = ρAL|￿ ||B | sin θ = IL|B | sin θ v￿ v￿ (9.2) where we used that the total charge is given by q = ρLA for a conductor of length L. Therefore the force is proportional to the current and the length, one can define a force per unit length as ￿ |F | ￿ = I |B | sin θ (9.3) L The direction and orientation of the force is given by exactly the same right-hand rule as before. Only that instead of the velocity we use the direction of the current. A very interesting observation is that the magnetic field allows to determine the sign of the charge density ρ responsible for the current. This is illustrated in fig.54 and known as the Hall effect. The same current can be produced by positive carriers moving in one direction along a wire or by negative ones moving in the opposite direction. However, the magnetic force on the wire is the same in both cases and for that purpose we do not need to know the sign of the carriers. On the other hand it means that the carriers would like to accumulate on one side of the wire and therefore if that side becomes positively charged the carriers are positive and if it becomes negatively charged they are negative. The experiment shows that the carriers are negative as we know since they are electrons. 9.2 Magnetic field created by a current 9.2.1 Ampere’s law We mentioned that an electric current creates a magnetic field as can be seen with an electromagnet, a coil through which a current circulates behaves as a magnet. The – 62 – Figure 52: This interesting demo shows that a current feels a force in the presence of a magnetic field. By switching the direction of the current the direction of the force changes and the aluminum bar rolls one way and then the other. Figure 53: A current is a charge density ρ moving with velocity ￿ through a conductor of v cross section A and length L. The charge contained in a volume |￿ |∆t A goes through the v cross section A during a time ∆t. This gives I = ρ|￿ |A. v intensity of the magnetic field is given by Ampere’s law. We mentioned that the flux of the magnetic field through a surface is zero. However given a vector field there is another important quantity known as the circulation. Given a closed path one multiplies the component of the magnetic field in the direction of the path times the displacement and sums over all portions of the path: ￿ ￿ Circulation around a path = |B |∆￿ cos θ (9.4) where ∆￿ is the displacement, θ the angle between the magnetic field and the direction of the path and the sum is over all portions in which we divide the path for convenience – 63 – Figure 54: The same current is produced by positive charges moving to the right as by negative charges moving to the left. In a magnetic field both feel a force as indicated meaning that different charge accumulates on the sides of the conductor depending on the sign of the carriers. This is the Hall effect and allows to determine the sign of the particles responsible for the current. of the calculation (see fig.55). It is exactly the same type of formula used to compute the work done by a force when you move and object from one place to another. Only that here we use the magnetic field and not the force in the computation (the magnetic force does no work anyway). What Ampere’s law says is that the circulation around any closed path is proportional to the current that pierces through any surface whose boundary is the given path. ￿ ￿ |B |∆￿ cos θ = µ0 I (9.5) where the constant µ0 is µ0 = 4π 10−7 Tm A (9.6) 9.2.2 Displacement current Maxwell realized that Ampere’s law contains an ambiguity and should actually be amended to be true in all cases. Indeed, the circulation of the current around a closed path is equal to the current piercing a surface whose boundary is the loop. However there are many such surfaces. In fig. 56 we see two surfaces ending in the circular path. One (S1 ) is just a disk whereas S2 is a dome-shaped surface. By charge conservation, if the current going through S1 is not the same as the one going through S2 , charge has to accumulate in the volume between the two surfaces. For example, if we put – 64 – Figure 55: The circulation of the magnetic field is defined in a similar way as the work of a force along a path. In fact the circulation along a path can be defined for any vector field. a capacitor as in the figure, we can charge the capacitor and no current would go through S2 whereas, at the same time, current I is going through S1 . This makes Ampere’s law unclear since which surface should we use?. Here Maxwell realized that a time dependent electric field would be piercing surface S2 since the capacitor is being charged and an increasing electric field exists between the two plates of the capacitor. So he proposed that the circulation of the electric field should be equal to the current plus the variation of the electric flux through the surface. He also realized that this had profound consequences, a varying electric field creates a magnetic field. We are going to see that a varying magnetic field creates an electric field and so on. This process gives rises to a wave that propagates in space. In this way Maxwell predicted the existence of radio waves and later also suggested that light could be such an electromagnetic wave. It is important to study this reasoning in detail. It illustrates perfectly the way that theoretical physics works. Starting from Ampere’s law that had been verified – 65 – experimentally, Maxwell realized an ambiguity, corrected it therefore predicting the existence of a new phenomenon, electromagnetic waves, which was later verified by Hertz. It is very instructive and a great scientific achievement. Figure 56: The current through two surfaces, S1 and S2 is different but through S2 there is a time varying electric flux. 9.2.3 Magnetic field of a wire and a solenoid (coil) We can use Ampere’s law to compute the magnetic field surrounding a cable. For a straight cable the magnetic field goes around as indicated in fig.57. Using Ampere’s law for a circular path around the cable we get circulation = 2π rB = µ0 I We then have (9.7) µ0 I (9.8) 2π r We can create a stronger magnetic field by putting several cables such that their magnetic fields add up. In fact we do not need to put different wires, we can take one ￿ |B | = – 66 – wire and form a coil as in fig.58. In that case the magnetic field is non-zero inside and outside is very small (if the coil is very long). Using a path as in the figure we get from Ampere’s law: BL = µ0 IN (9.9) where N is the number of turns of the coil and L is its length. Therefore, the magnetic field inside the coil is N ￿ B = µ0 I z ˆ (9.10) L Figure 57: Magnetic field produced by straight wire. – 67 – Figure 58: Using Ampere’s law to find the magnetic field inside a coil. – 68 – ...
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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.

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