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Notes On Electricity - 2

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Published in: Physics
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Notes on Physics

Aritra D / Kolkata

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  1. Electricity Introduction Electric fields are only part of the electric story; you have other concepts to take into account. You're dealing with forces in electricity, so you have to address the idea of potential energy, or the energy stored in an object or system. Mixing force and potential energy is a natural fit; for example, when you lift a weight in a gravitational field, you end up with potential energy, the energy stored in the object because of its new position: PE = mghf— mgho where m represents mass, g represents the acceleration due to gravity, hfrepresents the final height, and ho represents the initial height. Because a force acts on the charges in an electric field, you can also speak about potential energy in electric fields. Such potential energy is electric potential energy, and a change in electric potential energy creates a new quantity called voltage, or the driving force of electric current. This chapter shows that the electrical force on a charge q leads to electrical potential energy, denoted by U. The amount of electric potential energy equals the product of q and a quantity V (italicized) called the electrical potential, whose unit is the volt (V, not italicized). Just as for gravity, where only differences in the gravitational potential are meaningful, so for electricity, only differences in the electrical potential are meaningful. Sometimes electrical potential energy is called merely electrical energy. Electric Potential and Potential Difference When a charged particle is made to move in an electrostatic field in a direction opposite to the direction of the field, work is done by an external agency. This work is stored as potential energy of charge in accordance with the law of conservation of energy. So we can say that an electric charge placed at a point in an electric field has potential energy, which is a function of its position. We can visualize the potential energy of charge in the field as a scalar function of position and for a unit charge call it potential. It means that different points in an electric field would be at different potentials. And if a positively charged particle is placed in an electric field, it will tend to move from higher to lower potential to minimize its potential energy. In the next lesson, you will learn how the concept of potential difference leads to flow of current in electric circuits. The electric potential at any point in an electric field is equal to the work done against the electric force in moving a unit positive charge from outside the electric field to that point. It may also be defined as the amount of electric potential energy per unit charge when a positive test charge is brought in from infinity. Electric potential is a scalar quantity, as it is related to work done. There are two ways to consider electrical potential. The first is from the action-at-a-distance view point, the electrical potential is obtained as a sum of the contributions from every electric charge, with the potential at infinity conventionally taken to be zero. This requires a
  2. complete knowledge of the positions and magnitudes of every charge. To obtain the electrical potential difference between two points thus requires the potential at both points. The second is from the field viewpoint, the electrical potential difference between those two points is obtained as an integral over the electric field along some path between them. The field viewpoint is particularly useful in describing real electrical conductors in equilibrium, here typically the detailed surface charge distribution 'Vs is unknown, but the potential over the conductor is known. Conductors in equilibrium are characterized by a fixed value for the electrical potential, which is the same over the surface and even the volume of the conductor. (Thus, they are called equipotential surfaces. ) You already know something about electrical potential. The terminals of a battery or the prongs of an electrical outlet provide a difference in electrical potential. Voltage and electrical potential are used interchangeably. The electrical force on a charge q can be written as the product of q with the electric field E: Let us develop a similar relationship between the electrical energy U of a charge q as the product of q with the electrical potential V: U = qV (electrical energy from electrical potential) (ii) From (ii), the electrical potential V has units of J/C, which defines the volt (V). Just as for gravity, only differences in electrical potential energy and electrical potential have meaning. Usually, such differences are measured with respect to "ground," which is taken to be at zero potential. The electrical potential concept has the same advantage as the electric field concept: it is a property of position and is independent of the test charge. The potential at a point is taken positive when work is done against the field by a positive charge but negative when work is done by the electric field in moving the unit positive charge from infinity to the point in the field. Consider two points A and B in an electric field as shown in Figure 1. If a test charge qo is moved from point A to point B along any path by an external force, the amount of work done by the external force is given by WAB = qo (VB — VA) (iii) Thus, potential difference between points A and B will be w AB (iv) VAB = VB — VA = Here VA and VB are potentials at points A and B, respectively. A potential difference is said to exist between two points in an electric field, if work is done against the electric force in moving a positive test charge from one point to the other.
  3. Note that this work is independent of the path. (For this reason, the electric field is said to be a conservative field). The Sl unit of potential and potential difference is volt: 1 volt = 1 joule/ 1 coulomb 4 1 Figure 1: The work done in moving a test charge from one point to another in an electric field is independent of the path followed. If one joule of work is done in taking a test charge of one coulomb from one point to the other in an electric field, the potential difference between these points is said to be one volt. If one joule of work is done in bringing a test charge of one coulomb from infinity to a point in the field, the potential at that point is one volt. Potential at a point is not a unique quantity as its value depends on our choice of zero potential energy (infinity). However, the potential difference between two points in a stationary field will have a unique value. Motion Through an Electric Field Change in Electric Potential. If we move from an initial point i to a second point fin the electric field of a charged object, the electric potential changes by (v) If we move a particle with charge q from i to f, then, from Equation (ii), the potential energy of the system changes by AU = qAV= q(Vf Vi) (vi) The change can be positive or negative, depending on the signs of q and AV. It can also be zero, if there is no change in potential from i to f (the points have the same value of potential). Because the electric force is conservative, the change in potential energy !U between i and f is the same for all paths between those points (it is path independent). Work by the Field. We can relate the potential energy change AU to the work W done by the electric force as the particle moves conservative force: W = — AU (work, conservative force) from i to f by applying the general relation for a (vii)
  4. Next, we can relate that work to the change in the potential by substituting from Equation (vi): W=-AU =-qAV = -q(Vf Vi) (viii) Up until now, we have always attributed work to a force but here can also say that W is the work done on the particle by the electric field (because it, of course, produces the force). The work can be positive, negative, or zero. Because AU between any two points is path independent, so is the work W done by the field. Conservation of Energy. If a charged particle moves through an electric field with no force acting on it other than the electric force due to the field, then the mechanical energy is conserved. Let's assume that we can assign the electric potential energy to the particle alone. Then we can write the conservation of mechanical energy of the particle that moves from point i to point f as: or AK = — AU Substituting Equation (vi), ... (ix) we find a very useful equation for the change in the particle's kinetic energy as a result of the particle moving through a potential difference: AK q AV=- q(Vf- Vi) (xi) Work by an Applied Force. If some force in addition to the electric force acts on the particle, we say that the additional force is an applied force or external force, which is often attributed to an external agent. Such an applied force can do work on the particle, but the force may not be conservative and thus, in general, we cannot associate a potential energy with it. We account for that work Wapp by modifying Equation (ix): (initial energy) + (work by applied force) = (final energy) (xii) or I-Ji+ Ki+ Wapp = Uf+ Kf Rearranging and substituting from Equation (vi), we can also write this as (xiii) AK=-AU+ wapp =-qAV+ wapp The work by the applied force can be positive, negative, or zero, and thus the energy of the system can increase, decrease, or remain the same. In the special case where the particle is stationary before and after the move, the kinetic energy terms in Equations (xii) and (xiii) are zero and we have Wapp = q AV (for 16 = Kf) (xiv)
  5. In this special case, the work Wapp involves the motion of the particle through the potential difference AV and not a change in the particle's kinetic energy. By comparing Equations (viii) and (xiv), we see that in this special case, the work by the applied force is the negative of the work by the field: app — W (for Ki = Kf) (xv) Electron-volts. In atomic and subatomic physics, energy measures in the Sl unit of joules often require awkward powers of ten. A more convenient (but non-Sl unit) is the electron-volt (eV), which is defined to be equal to the work required to move a single elementary charge e (such as that of an electron or proton) through a potential difference !V of exactly one volt. From Equation (viii), we see that the magnitude of this work is q AV. Thus, 1 ev = e(1V) = (1.602 x 10-19 J/c) = 1.602 x 10-19 J (xvi) Equipotential Surfaces Adjacent points that have the same electric potential form an equipotential surface, which can be either an imaginary surface or a real, physical surface. No net work W is done on a charged particle by an electric field when the particle moves between two points i and fon the same equipotential surface. This follows from Equation (viii), which tells us that W must be zero if Vf= Vi. Because of the path independence of work (and thus of potential energy and potential), W = 0 for any path connecting points i and f on a given equipotential surface regardless of whether that path lies entirely on that surface. Figure 2 shows a family of equipotential surfaces associated with the electric field due to some distribution of charges. this geth an an surface. 11 No d:ne *Jcr.g tiis pat' thatretums to the •.rre u-fa:e. Egg] dcne *Jang these *ths tet'öZl the sa-ne surßcs. V, Figure 2. Portions of four equipotential surfaces at electric potentials VI = 100 V, = 80 V, 1/3 = 60 V and V4 = 40 V. Four paths along which a test charge may move are shown. Two electric field lines are also indicated. Capacitance
  6. A capacitor stores charge by holding charges separate so that they attract each other but don't have a way to go from one plate to the other by themselves. How much charge is stored? That depends on the capacitance, C, of the capacitor. The amount of charge that appears on both plates of the capacitor is equal (the charges are opposite in sign) and depends on the voltage between those plates, as given by the following equation, where C is the capacitance: (xvii) q = CV where q is an electric charge. Capacitance is defined as the ratio between the charge on either of the conductors and the potential difference between them. It is a measure of the capability of a capacitor to store charge. For a parallel plate capacitor, the electric field, E, equals the following: (xviii) where E is a constant and A represents the area of the plates. And the voltage between the plates separated by a distance s is (xix) Therefore, V = qs/EOA (xx) Because q = CV, you can solve the preceding equation for q / v to get C=q/V= A/s (xxi) The equation C = q / V = Eo A/ s allows you to find the capacitance for a parallel plate capacitor whose plates each have area A and are a distance s apart. The MKS unit for capacitance is Coulombs per volt, also called the Farad, F. The capacitance is one farad, if a charge of one coulomb creates a potential difference of one volt: 1 farad = Icoloumb / 1 volt (xxii) Types of Capacitors There are three common varieties of capacitors in commercial use. 1. Paper capacitor: Several large thin sheets of paraffin impregnated paper or mylar are cut in proper size (rectangular). Several sheets of metallic foils are also cut to the same size. These
  7. are spread one over the other alternately. The outer sheet is mylar, then over it a sheet of metal foil, again over it a sheet of mylar and then a sheet of metal foil and so on. The entire system is then rolled in the form of a cylinder to form a small device. 2. Metal plate capacitors: A large number of metals are alternately joined to two metal rods as shown in Fig.16.12 (b). The entire plate system is immersed in silicon oil which works as dielectric material between the plates. High voltage capacitors are usually of this type. Variable capacitors of micro farad capacitance are usually of this type and use air as dielectric. One set of plates is fixed and the other set is movable. The movable plates, when rotated, change their effective area, thereby changing the capacitance of the system. You might see such capacitors in a radio receiver. Variable capacitance helps in tuning to different radio stations. 3. Electrolytic capacitor: An electrolytic capacitor is shown Figure 3 (c). A metal foil is rolled in the shape of a cylinder with increasing diameter so that there is always a space between one surface and the other. The system is immersed in an electrolyte in the form of a solution. This solution is conducting because of ions in the solution. A voltage is applied between the electrolyte and the metallic foil. Because of the conducting nature of the electrolyte, a thin layer of metal oxide, which is an insulator, is formed on the foil. The oxide layer works as dielectric material. Since the dielectric layer is extremely thin, the system provides a very high value of capacitance. It is important in this type of capacitor to mark the positive and negative terminals. A wrong connection of positive and negative terminals removes the oxide layer. (The capacitor then starts conducting.) This type of capacitor is used in storing large amount of charge at low voltage. rte elb:: F arr Figure 3: Different types electrolytic capacitor. Dielectric of capacitors: (a) paper capacitor, (b) variable capacitor and (c) Most capacitors don't depend on just air between the plates — they use a dielectric between the plates. A dielectric is a semi-insulating material that increases how much charge the capacitor can hold by its dielectric constant, K. So, when the space between the plates of a parallel plate capacitor is filled with a dielectric of dielectric constant K, the capacitance increases to C = KEO A/s ... (xxiii)
  8. For example, the dielectric constant of mica (a mineral commonly used in capacitors) is about 5.4, so it increases the capacitance of a capacitor to 5.4 times that of the same capacitor with a vacuum between the plates, because a vacuum has a dielectric constant of 1.0. Effects of Electric Current When an electric current flows through a circuit, it causes the following effects. Heating Effect of Electric Current: When an electric current is sent through a wire, electrical energy gets converted into heat energy. Therefore, the wire gets hot. This effect is known as the heating effect of electric current. Some of the home appliances that work on the property of the heating effect of electric current are electric room heater, electric iron, toaster, hair dryer, electric stove, immersion water heater, coffee maker, electric rice cooker and so on. These appliances have coils of wire, known as heating elements, which produce heat. As current flows through these electrical appliances, a huge amount of heat is produced. Different appliances have different types of heating elements. Magnetic Effect of Electric Current: In a crane, the end has a strong electromagnet to lift heavy loads. The electromagnets work on the basis of magnetic effect of electric current. When the electric current flows through conducting wires, the wires become a magnet temporarily till the current is passed. This property of electric current is used to separate metals from junks. Doctors use small electromagnets to take out small pieces of magnetic materials that have fallen into our eyes. Chemical Effect of Electric Current: When electric current is passed through a conducting solution, such as salt solution, some chemical reaction takes place. For example, when electric current is passed through the solution of a metal salt, such as solution of copper sulphate, metal gets deposited at the negative pole, because metal is positively charged. G) Ceth:de Cappe-• sulph3te Figure 4: Electric Circuit showing the chemical effect of electric current Electroplating is one of the chemical effects of electric current. Electroplating is a chemical process using which a metal is coated with a layer of another desired metal. Electroplating is done to make the metals shiny and appealing. For example, wheel rims, handle of cycle, etc. are made shiny by the method of electroplating.
  9. Mechanical Effect of Electric Current: When we press the switch ON, an electric current is sent to the motors in some appliances, including fans, grinder and exhausting fan. This electric supply makes it spin. This is called mechanical effect of electric current. Distribution of Electricity to Homes Electricity produced in a power house is transferred to our houses. It is supplied for usage through electric wires at your home from huge power stations. Electricity supplied by a.c. mains is used at homes. Electric wires carrying high power electricity hanging through large poles can be easily seen at any roadside. In our houses, the cables are taken to the main board where the electric meter is fixed. Electric Meter: The electric meter measures the quantity of electricity used. The meter measures electric energy in kilowatt-hour (kWh). 1 kilowatt-hour is called as 1 unit. Socket: From the meter, electricity is sent to sockets by conducting cables. Sockets can have 2 or 3 outlets. In a 3-outlet socket, the third big hole is for saving us from shock and thus known as earth connector. Electric Switch: A switch is an electrical device, which is used to stop or start the flow of electricity in an appliance to work. The outer part of the switch is made up of insulator material such as Bakelite, ebonite, etc. Switch is connected to the live electric wire. When we put switch in an ON position, the current will flow as the circuit is closed. If a switch is in OFF position, then supply of current stops. Electric Fuse: The electric fuse works on the principle of the heating effect of electric current. An electric fuse is a wire made from some special materials which melt quickly and break when large electric currents are passed through them. In all buildings, fuses are inserted in all electrical circuits. There is a maximum limit on the current, which can safely flow through a circuit. As the current increases beyond a limit, the wire in the electric fuse melts and breaks off. The fuse is then said to have blown off. The circuit is broken and current stops flowing through it. A fuse is thus a safety device which prevents damage to electrical circuits and possible fires. Miniature Circuit Breakers (MCBs): When a fuse wire breaks due to excessive current, every time, we need to replace it by another one. Therefore, instead of fuses, MCBs are used nowadays. MCBs are switches that turn off automatically when there is an excessive current. After solving the problem in the circuit, we can turn ON the switch and then the current flows usually.

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