lecture2 - 2. Lecture 2 2.1 Electric charge In the same way...

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Unformatted text preview: 2. Lecture 2 2.1 Electric charge In the same way that gravity describes the interaction of masses, electrostatics describes the interaction of electric charges. Notice that we say electrostatics because this applies to static charges. If charges move they generate a magnetic field. This can be ignored if the charges move slowly compared with the speed of light except if we have a large number of charges moving together in the same direction as in an electric cable. We will look at that later, for the moment we concentrate in static charges (or moving slowly). Before continuing however one might wonder if the law of gravity might not need to be amended and perhaps masses also interact differently if they move fast. This is actually true and it is described by Einstein’s theory of general relativity. Going back then to electric charges, a difference with gravity is that in this case the force can be repulsive as well as attractive. In fact electric charge can be positive or negative, charges of opposite sign attract and those of the same sign repel. Other than that, the law for the force is similar as established by Coulomb’s law: 1 q1 q2 ￿ |F | = (2.1) 4πε0 r2 Namely the magnitude of the force is proportional to the product of the charges and inversely proportional to the square of the distance. 2 1 The constant that replaces Newton’s constant is 4πε0 = 9 × 109 N m2 . This values C assumes that we measure the electric charge in Coulombs (C). In atomic and nuclear physics sometimes other units of charge are used so that the constant is just one. To get an idea of how much a Coulomb is we can consider the minimal unit of charge which is the charge of the electron e = −1.6 × 10−19 C . The proton, has the same but opposite charge. It is still a mystery why the charge of the proton and electron are exactly opposite but that implies that atoms are exactly neutral since they have the same number of protons (which are in the nucleus) and electrons (which orbit the nucleus). Perhaps it should be pointed out that the proton is made out of quarks called u and d that have charges +2/3e and −1/3e respectively. In any case we see that a large number of electrons are needed to make a charge of a Coulomb. However, the Avogadro number NA = 6 × 1023 is much larger. Since the Avogadro number is the number of atoms in a mol of a substance the means that we have available that number of electrons. For example one gram of Hydrogen (which is one mol) has NA = 6 × 1023 atoms each of which has one proton and one electron. However large charges are not usually obtained because, as mentioned before atoms are neutral. This is because the electric interaction is large and it would cost a large amount of energy to split positive and negative charges. –9– + + q1 q2 + − q1 q2 − − q1 q2 Figure 3: Two charges can attract or repel each other depending on their sign. Nevertheless, a small amount of charge can be created, one example is by rubbing two materials. Sometimes one of them gets charged and allows us to check the laws of attraction and repulsion. To understand more about charge we need to know that it is conserved. Experimentally it has been observed that the total charge of an isolated system is always the same. One can create charge but only of opposite signs in such a way that the total is always the same. So if you rub two objects and one is charged positive the other will be charged negative. Another important property is that of materials. Certain material such as metals are conductors, which means that electric charges can move freely inside them. Insulators on the other hand do not allow the motion of charge. These are the most common types, other materials such as semiconductors, superconductors etc. are more rare but very important in technological applications as we all know (electronics is based on semiconductors for example). If a metal is charged, since equal charges repel each other, the charges will try to be as far of each other as possible and will migrate to the surface. In fig.4 we see for example that if we approach a conducting sphere with a negatively charged rod, positive charge will be attracted close to the rod and negative charge well be repelled. However the total charge of the sphere should be zero if it was so initially. A different situation is if we connect the sphere to ground. This means that we run a conducting cable into the ground basically connecting the sphere to the Earth which for this purpose can be thought of as an unlimited reservoir – 10 – of charge. In that case the negative charge will be repelled all the way to the ground. If we then disconnect the sphere and afterward remove the rod, the sphere will acquire a charge. The process is summarized in fig.5. This phenomenon is call induction and can be used to generate relatively large amounts of charge. An example is the machine demonstrated in class (unfortunately not very successfully) and which can be seen in fig. 6. + − − − − − − − − −− −− + + + + + + + + + + − −− −− −− Figure 4: A negatively charged rod attracts positive charges and repels negative ones. The total charge of the sphere however remains constant since it is insulated The idea of connecting something to ground is extremely important and when using an electrical device an important point is if it is appropriately connected to ground. One hand connecting the chassis (metallic case) to ground is a safety precaution against accidentally connecting it to a power line. It also avoids static electricity that can damage electronic circuits. Although connecting to ground literally means a connection to a conductor embedded in the soil, sometimes this is not practical (for example cell phones etc.) and then the “ground” refers to a common connection of electrical parts to the chassis. This gives a stable common reference to all circuits. This is particularly important in sensitive electronic devices since this common reference gives them stability, otherwise they can function erratically and can be a common reason for – 11 – + + + + + ++ + + + + + + + + ++ ++ + + + ++ + − −− − − + ++ ++ + + + + + + +++ − −− − Figure 5: If the sphere is connected to ground we can induced a charge in it by using the procedure in the figure. Be sure you understand what happens in each step. malfunctioning (for example, you create your own circuit to plug to a computer port but forget to connect the ground of your circuit to the computer ground either through the port or directly to the chassis). Another interesting point is what happens if you have a charged conducting sphere and you touch it with another conducting one which is not charged. Since they are both conducting the charge will distribute among them. However, which one gets more charge? If the spheres have the same radius, by symmetry the total charge is distributed equally. However if one is bigger, the charges on it can be further apart which they prefer because the force between charges of the same sign is repulsive. What this means is that the sphere of larger radius gets more charge. In fact will see later that the charge it gets is proportional to its radius. – 12 – Figure 6: Machines used to generate static electricity. The first and last one are Wimshurst machines. In the middle is a small Van der Graaf generator. Van der Graaf made huge generators to accelerate atomic particles which can still be seen in the Boston museum of Science. 2.2 Electric field Although initially one might think that electricity describes only forces between charges, simple experiments show that electromagnetic waves propagate from one place to another. Those include radio waves, light etc. A simple demonstration of this is when we create sparks in class and that was detected by the AM radio receiver which generated a noise with each spark. All this suggests that there is a form of energy that exist independently of the charges and leads us to the idea of electric field. Charges are sources for the electric field but electric fields can exist independently of charges. We ￿ denote the electric field as E . It is a vector and it has the property that if you put a charge q in it, the charge experiences a force: ￿ ￿ F = qE (2.2) ￿ The direction of the force is parallel to E although it can have opposite orientation if the charge is negative. It also follows that the force is proportional to the charge as can be verified experimentally. Furthermore we find that we reproduce Coulomb’s law if a charge q1 generates an electric field ￿ E= 1 q1 r ˆ 4πε0 r2 (2.3) where r is a radial unit vector pointing away from the charge. An important principle ˆ that can be verified experimentally is the principle of superposition, if several sources ￿￿ ￿ produce fields E1 , E2 , . . . , En then the total electric field is ￿ ￿ ￿ ￿ E = E1 + E2 + . . . + En (2.4) This implies the principle of superposition for the forces acting on an object namely ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ F = q E = q E1 + q E2 + . . . + q En = F1 + F2 + . . . + Fn – 13 – (2.5) It is important that we add the forces and also the electric fields as vectors. Remember the rule of addition of vectors as illustrated in fig.7. Notice that vectors are mathematical entities with properties independent of what they represent. The easiest way to think about them is to think that they represent displacement from one place to another. Everybody is able to figure out where you are going to end up if you move lets say 30 meters in certain given direction, for example North and then 20 meters in another, for example South-West. This is addition of vectors and the same principle can be applied to the vectors represent electric fields, forces, velocities etc. But always add vectors that represent the same thing! By the way, we emphasized that every quantity has a unit and we see that electric field should be measured in N/C , where N is Newton, a unit of force and C a unit of charge. F 2 F 3 F 1 F=F1 + F2 + F3 Figure 7: Vector are added in the same way you add displacements form one place to another. Be sure you have no confusion on how to add vectors The concept of electric field is very useful but in principle it looks as a mess to draw. Indeed you have to draw an arrow at each point in space!. For that reason people found other way to represent the electric field and introduced the concept of “lines of electric field”. What you do is to draw lines in such way that they are always tangent to the electric field. This gives you the direction. Then one draws more lines where the field is more intense. This is not a precise representation but gives a pictorial idea of how the electric field actually looks like. The simplest case is that of a single charge that we draw in figures 8 and 9. Notice that the Electric field points away from a positive charge and toward a negative one. This is because a small positive probe charge will be repelled by the positive charge and be attracted by the negative one. – 14 – + Figure 8: A positive charge generates an electric field pointing away from it. – 15 – − Figure 9: A negative charge generates an Electric field pointing toward it. – 16 – ...
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