Lecture4 - 4 Lecture 4 4.1 More on electrostatic potential From formula(3.12 we see that V cannot have abrupt jumps otherwise the electric field

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Unformatted text preview: 4. Lecture 4 4.1 More on electrostatic potential From formula (3.12) we see that V cannot have abrupt jumps otherwise the electric field would be infinite. One example is the potential of a charged sphere which is constant inside and decreases as 1/r outside. The electric field on the other hand has a jump, it is zero inside and has a finite value on the surface. We plot this in figs.16 and 17. 1 0.8 V(r) 0.6 0.4 0.2 0 2 4 6 8 10 R Figure 16: Dependence of the electric potential with the distance to a center of a charged sphere. An interesting situation that we have already discussed is what happens if we touch two charged spheres of different radii, how is the total charge distributed between the two?. Now we know that it distributes so that the potential is constant. In a first approximation we can ignore the presence of the other sphere and say that the potential on the surface of each sphere is given by V1 = 1 Q1 , 4πε0 R1 V2 = 1 Q2 4πε0 R2 (4.1) where Q1,2 and R1,2 are the charges and radii of each sphere respectively. If the potentials are equal (V1 = V2 ) then we evidently have Q1 Q2 = R1 R2 – 25 – (4.2) 0.5 0.4 0.3 E(r) 0.2 0.1 0 2 4 6 8 10 R Figure 17: Dependence of the electric field with the distance to a center of a charged sphere. implying that the smallest sphere has less charge, namely the charge is proportional to the radius as mentioned before. Naively or pictorially speaking the charges prefer to be in the large sphere since they repel each other and can then be further apart. However one important observation is that the electric field is larger in the surface of the smaller sphere. Indeed we have ￿ |E1 | = 1 Q1 , 2 4πε0 R1 ￿ |E2 | = 1 Q2 , 2 4πε0 R2 (4.3) From here, and eq.(4.2) it is evident that ￿ ￿ | E 1 | R1 = | E 2 | R2 . (4.4) ￿ We see that if R1 is very small then |E1 | is very large. Notice this is because, although Q1 is smaller, we can get closer to the center of the small sphere. An extreme case is when we have a sharp point which can be thought as a sphere of almost zero size. Near it the electric field will be extremely large ionizing the air and allowing charge to flow out of the conductor. This is the principle of the lightning rod invented by Benjamin Franklin. Lightning is essentially a giant spark. The lightning rod ionizes the air around making it conductive and favoring the initiation of the spark. Therefore lightning is more likely to strike on the lightning rod than on the structure that it protects. Another important observation is that, since the electric field inside a conductor is zero, it will still be zero if we carve a hole inside it (fig.19). This means that – 26 – R2 R1 Q 1 Q2 Figure 18: Distribution of charges between two conducting sphere of different radii and in contact with each other. conductors act as shields for electric fields. An important application is the so called Faraday cage. Inside a metallic cage no electric field penetrates and therefore it is isolated from electromagnetic waves, namely radios, cell phones etc. do not work. 4.2 Electric flux One interesting concept that can be defined for a vector field such as the electric field is that of flux through a surface. The idea originates from considering the motion of water. If water is moving at velocity ￿ then the flow of water (usually given in liters v per second, or gallons per minute) through a surface of area A is given by flow = v⊥ A = |￿ |A cos θ v (4.5) where v⊥ = |￿ | cos θ is the component of the velocity perpendicular to the surface. Here v θ is the angle between the normal and the velocity. To see why this is so consider figure 20. During a time interval ∆t it is clear that all the fluid contained in the volume A|￿ | cos θ ∆t will go through the area A and hence the result. v – 27 – − − E=0 − − + + + − − + E=0 − + − + − + − + − + + − + − Figure 19: In a region with no charge and surrounded by a conductor the electric field is zero. By analogy1 , given a surface we define the electric flux through it as the area of the surface times the value of the component of the electric field perpendicular to the ￿ surface. The easiest case is when the surface is everywhere perpendicular to E in which ￿ case we just multiply area times |E |. For example for a single charge, we can take as a surface the sphere of radius R concentric with the charge (fig.21). In that case we have ￿ that the surface is perpendicular to E and the flux is flux = 4π R2 1Q Q = 2 4πε0 R ε0 (4.6) namely it is independent of the radius R !. Moreover, consider a cone that determines a surface as the one in figure 22, namely a truncated cone where the base and the top are spherical. Notice that the areas are related by A1 A2 =2 2 R1 R2 1 (4.7) In the case of the electric field the situation is static, there no fluid moving but it is a useful analogy – 28 – θ v A |v| cos θ ∆ t Figure 20: Flow of water through a cross section area A. because the area of an object scales as the square of the (linear) size. Along the laterals of the truncated cone the flux is zero since it is parallel to the electric field. Through the base and the top the flux has opposite signs since it is entering through one and exiting through the other. The total flux is flux = 1Q 1Q A− A =0 21 22 4πε0 R1 4πε0 R2 (4.8) It vanishes in view of eq.(4.7). Therefore the flux through this surface which does not surround the charge is zero. With some imagination one can think of constructing any surface with very small blocks of this (truncated) conical shape and the result will be the same. If the surface does not surround the charge the flux is zero and if it does Q then the flux is ε0 . The superposition principle also applies to the flux, so if the surface surrounds several charges the flux is given by the total charge surrounded. This is Q called Gauss’s theorem, the flux through a closed surface is given by ε0 where Q is the total charge surrounded by the surface. It is a very powerful theorem since one can take a surface very far from the charges and knowing the electric field there is enough information to know how much charge we enclose. – 29 – Q R Figure 21: Computing the flux of the electric field produced by a charge through a concentric sphere of radius R. One simple example where we can use the theorem is for a flat surface with a charge density σ . Namely, if we cut a region of area A of the surface, the charge contained is σ A. Looking at fig.23 and by symmetry the electric field will be uniform and pointing away from the plane. Taking different surfaces and computing the flux we see that ￿ 2|E | A = σA ε0 ⇒ ￿ |E | = σ 2 ε0 If the plane is (x, y ), that is normal to z then we have an electric field ˆ ￿ σ z if z > 0 ˆ ￿ E = 2ε 0 σ − 2ε0 z if z < 0 ˆ (4.9) (4.10) In fact you can show using Gauss theorem that the electric field has to be independent from the distance to the plane. Try to see why this is so using a cylindrical surface which does not intersect the plane. – 30 – R2 , A 2 R1 , A 1 Q Figure 22: For the truncated cone in the figure, the total flux is zero since the flux incoming at the bottom is equal to the outgoing at the top. Figure 23: The electric field created by a plane can be computed using Gauss law. 4.3 Electric current As we explained, electrostatic forces are responsible for chemistry, explain why solids are solids whereas liquids are liquids etc. It has practical applications in ink-jet printers, electrophoresis, cathode tubes, particle accelerators, etc. However, by far the most common use of electricity is through the use of electric currents and circuits. The idea is that if we have a constant electric field in a conductor, this will produce a motion of charges until the electric field vanishes inside. However the idea emerges – 31 – that if we can remove the charges from one side and put it back in the other then we can have a continuous motion of charge from one place to another. This “removal” of charge should be done against the potential and therefore requires energy. The device that produces such effect is known as a battery. Usually the energy in a battery comes from a chemical reaction inside it. Such chemical reactions are capable of moving charges across a potential difference of around 1 V ( the usual battery if 1.5 V). A simple comment is how does then the 9V battery work?. Well, by opening up one it is easy to see that there are 6 “elements” inside it. That essentially means six 1.5V batteries connected in series, that is one after the other. We see that the potential just adds as actually follows from the properties of V we discussed before. Figure 24: Typical chemical batteries have an emf of 1.5V. A 9V battery consists of 6 of those in series, that is one after the other. If there were no resistance to the flow of charge, the electrons would accelerate indefinitely inside the conductor. As it were, there is an effective maximum velocity and therefore, after a short time, if a conductor is connected to a battery a steady flow of charge will be present. Such flow of charge is known as a current and is measured in Amperes (or Amps). A current of 1A means that one Coulomb of charge is going through a section of the conductor every second 1A = 1 C . Experimentally it turns out s that the current inside a conductor is proportional to the potential difference or voltage across it. This is called Ohm’s law and reads: V = IR – 32 – (4.11) where V is the voltage across a conductor, I is the current and R is the resistance. The resistance in measured in Ohms (Ω) where 1Ω = 1 V . So a potential difference of A 1V across a resistor of 1 Ω creates a current of 1A. Remember also that the potential always decreases along the direction of the current. Examples of resistors are shown in figure 26. Typically such resistors are color coded. By reading the colors one can figure out the resistance. digit color multiplier silver 0.01 gold 0.1 0 black 1 1 brown 10 2 red 100 3 orange 1K 4 yellow 10K 5 green 100K 6 blue 1M 7 violet 10M 8 gray 9 white The way it works is explained in fig.27 where we see that the value is written with the first three stripes. The first two are read as a number using the table. The third one is a multiplier whose value is also taken from the table. For example red black orange is read as 20K Ω = 20, 000Ω. – 33 – I R + − V Figure 25: Simple circuit with a battery and a resistor. By Ohm’s law we have V = I R. Figure 26: Examples of commonly used resistors. – 34 – multiplier st 1 digit 2 nd digit Figure 27: The first three stripes indicate the value of the resistance according to the color code table. The other ones are related to tolerance and reliability. – 35 – ...
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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-West Lafayette.

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