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lecture8 - 8 Lecture 8 8.1 Capacitors in series and...

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Unformatted text preview: 8. Lecture 8 8.1 Capacitors in series and parallel In the same way that one can analyze resistors in series and parallel one can understand what happens for capacitors in series and parallel. Notice that whereas Ohm’s law is ∆V = IR we have Q = C ∆V . So the potential difference or voltage ∆V is proportional to R but inversely proportional to C . As a consequence, and working with the circuits in fig.45 one finds that Series: 1 C = 1 C1 + 1 C2 + 1 C3 Parallel: C = C1 + C2 + C3 Try to derive these relations as an exercise. As a hint notice that when the capacitors are in series, the conductors in the middle are neutral so the charge in each capacitor is the same (see fig.45). +Q −Q +Q −Q +Q C1 C2 +Q −Q C3 −Q 1 1 C1 +Q 2 −Q 2 C2 +Q 3 −Q 3 C3 Figure 45: Capacitors in series and parallel. In parallel they add and in series their inverse add. Notice the difference with the case of resistors. – 55 – 8.2 Magnetic field Magnetism is a well-known phenomenon, for example the magnetic field of the Earth has provided orientation through the use of the compass for thousands of years. Magnetic fields can be generated by permanent magnets, electric currents and time dependent electric fields. In the case of the magnet the magnetic field looks like the one in fig.46 (see also the demo in fig. 48). Outside the magnet it is very similar to the one of a dipole electric field. But inside it is not. The lines of magnetic field close, they do not start or terminate at any point. Without knowing that, one might, at first sight, think that cutting in half a magnet will separate two opposite magnetic charges. Traditionally they are called the North and South pole. A pure North pole would have a magnetic field as the one in fig.47 However such object has never been observed in Nature. In particle physics there are theoretical indications that heavy particles with the properties of magnetic monopoles might exists. For that reason it is still an open question if monopoles (as they are called) exist or not. In any case if they do exist they are not common objects and we are not going to consider them here. The question naturally arises of what happens if we keep dividing the magnet, do we find an ”elementary magnet”?. The answer is yes, in fact the elementary magnet is our old friend the electron. Electrons not only have charge but they also behave as a tiny magnet. The superposition of the magnetic fields of all electrons creates the field of the magnet. Proton and neutrons are also tiny magnets but their strength is 2000 times smaller than the one of the electrons. An important consequence of the fact that there are no magnetic monopoles is that the flux of the magnetic field through a closed surface is always zero. This is because a theorem similar to Gauss’ theorem applies. The total flux is proportional to the total magnetic charge enclosed but since there is no magnetic charge, the flux is always zero, namely for each closed surface the same amount of magnetic flux comes in as it goes out. ￿ The magnetic field is a vector usually denoted as B . It can be detected by the effect it causes on charges. When a charge moves in a magnetic field it experiences a force equal to ￿ ￿ F = q E + q￿ × B v￿ (8.1) ￿ where ￿ is the velocity, q the charge and B the magnetic field. We included also an v ￿ electric field E for completeness. We need to explain what the operation ￿ × B is. It v￿ is known as a vector product and given two vectors it gives another vector: ￿ × B = |￿ ||B | sin θn v￿ v￿ ˆ – 56 – (8.2) Figure 46: Magnetic field of a permanent magnet. Outside is similar to an electric dipole but inside there are no charges where the lines end. They lines of magnetic field are actually closed. Its magnitude is the product of the modulus of ￿ , B and the sine of the angle between v￿ them that we denote as θ. Its orientation we denote by the unit vector n and is ˆ ￿ . We write that as perpendicular to both ￿ and B v n ⊥ ￿, ˆv ￿ n⊥B ˆ (8.3) Its orientation is given by the right-hand rule. If we move our fingers in a screw motion ￿ going from ￿ to B then our thumb points in the direction of the force. This is illustrated v in fig.50. Some important consequences are • If the particle is at rest then ￿ = 0 and there is no magnetic force. v ￿ • If the velocity is parallel to B then θ = 0 and also there is no force. • The magnetic force is always perpendicular to the velocity so it never does work!. – 57 – ? N Figure 47: A conjectured magnetic monopole will have all lines of magnetic field coming out. It has not yet been observed in Nature but theoretical ideas suggest it might be an actual but exotic particle. Figure 48: The lines of magnetic field can be made evident using iron filings which orient themselves parallel to the magnetic field. Also form the form of the Lorentz force we see that the magnetic field is measured Ns in units of Cm , such unit is known as a Tesla: 1T = 1 Ns Kg =1 Cm Cs (8.4) where we used 1N = 1 Kgm . To have an idea the magnetic field of the Earth is around s2 10−5 T but in the lab fields of 10T can be produced. Since the magnetic force is perpendicular to the velocity it will change its direction but not its magnitude. Therefore, if there are no other forces present, the kinetic energy – 58 – B v q F ￿ Figure 49: A particle moving in a magnetic field B experiences a force perpendicular to the ￿ . Its orientation is given by the right-hand rule. velocity and to B E = 1 mv 2 will remain constant, a consequence of the fact that the magnetic force does 2 no work. Furthermore, if the magnetic field is constant the magnitude of the force will be constant and the trajectory of the particle will be a circle in a plane perpendicular to the magnetic field. Indeed, in fig.49 we see that if the particle moves in circles then the force points always toward the center keeping the particle in its trajectory in the same way that we can tie an object at the end of a rope and make it move in circles. Using Newton’s law ￿ F = m￿ a (8.5) we can compute the the period of motion in a similar manner that we discussed at the beginning of the course for the Moon orbiting the Earth. the acceleration is in the radial direction, namely it is centripetal acceleration whose magnitude is |￿ | = a v2 r (8.6) as we remember from the mechanics course. Since the force point in the same direction we only need to equate their magnitudes: ￿ |F | = qvB = m giving v= qBr m – 59 – v2 r (8.7) (8.8) F B v Figure 50: Example of the right-hand rule. The force is the vector product of the velocity and the magnetic field and has the orientation indicated here. The period of motion is how long does it take for a particle to go around the circle. It is given by 2π r m 2π m ∆t = = 2π r = (8.9) v qBr qB amazingly it is independent of the radius. Namely if the radius is large the particle moves faster and takes the same time to go around. Notice that a peculiar property of this motion is that the particle can move in circles only in one direction. If one tries to move the particle around the circle in opposite direction then the force will point outward, it will not stay in the trajectory. Actually will again describe a circle in the “correct” direction. On the other hand the motion in the other direction is precisely what happens for a particle of opposite charge. Although rather simple this idea has had numerous applications. The most significant one perhaps is in particle accelerators where magnetic fields keep particles in circular trajectories while at the same time electric fields accelerate them at higher and higher energies. When these particles collide with fixed targets (or two opposite beams are brought into collision) new particles are created revealing the fundamental constituents of matter and their interactions. – 60 – F v B ￿ Figure 51: A free particle moving in a constant magnetic field B moves in circles since the magnetic force is oriented toward the center. – 61 – ...
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