Chapter 1 Electrokinetics PDF

Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Chapter 1 Electrokinetics : Charges and fields I. Introduction What is electrokinetics? Electr...

Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Chapter 1 Electrokinetics : Charges and fields I. Introduction What is electrokinetics? Electrokinetics is a term applied to all the laws of physicochemical and mechanical phenomena involving the transport of charges, the action of charged particles, and the effects of applied electric potential in order to allow a desired movement (in one direction or slowly reversible) of electric charge carriers in conductors forming closed electrical circuits. I.1. ELECTRIC CHARGES (symbol q, sometimes Q) Every physical phenomenon involves an “object” whose structure confers certain properties on the space around it. In the case of gravitation, the object is a mass. In electrostatics, the object is a charge, measured in coulomb (C) in the international system, however, in certain contexts, other units such as the ampere-hour (A h) are sometimes used. There are two types of electric charge distinguished by their positive or negative signs. Charges of the same sign repel each other, while those of opposite signs attract each other. In ordinary matter, there is a balance between positive and negative charges, known as electrical neutrality. One is called “positive” and is measured by a positive number, the other is called “negative” and is measured by a negative number. Fig 1: Electric field created by two charges of opposite signs. Any charge is a multiple of the elementary charge ( ) defined by: Atoms are made up of charged particles, namely: - Electrons: (e-) responsible for electrical conduction in metals: - Protons: (H+) : Two other properties of electric charge are essential to the electrical structure of matter: Academic year : 2024-2025 1  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Charge is conserved : The total electric charge in an isolated system, that is, the algebraic sum of the positive and negative charge present at any time, never changes;  Charge is quantized: The electric charges we find in nature come in units of one magnitude only, equal to the amount of charge carried by a single electron. These properties imply a quantity of charge and therefore a measurement of charge. We will now explain precisely how charge can be measured in terms of the force between charges located at a certain distance. II. ELECTROSTATICS Electrostatics is the branch of physics that studies phenomena created by static electric charges, i.e. the study of charges in their resting state. Electrification is the phenomenon of the appearance of an electric charge or the appearance of quantities of electricity on a body. Example: There is a simple experiment that anyone can do to feel an electrostatic force. All you have to do is rub a plastic ruler with a dry cloth and bring it close to small pieces of paper: this is electrification. We distinguish: - Point charges: assumed to be dimensionless, which is analogous to the material point hypothesis in mechanics? - Continuous charge distributions: the assumption of a macroscopic charge enables us to define an infinitesimal charge , to which we can apply the formulas established for the case of a point charge, before integrating on the distribution. In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. The charges densities are defined as follows: to which correspond respectively the infinitesimal charges II.1. Coulomb's law The interaction between electric charges at rest is described by Coulomb’s law: two stationary electric charges repel or attract one another with a force proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. Let two charges q and q′ be placed at M and M′ and spaced r (in meters). These charges can be positive or negative, but in the case of the figure, we'll assume they have the same sign. Academic year : 2024-2025 2  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Coulomb's law can be used to determine the force exerted by q on q′, or the force exerted by q′ on q, these two forces being equal and opposite, in accordance with principle of action and reaction. This law is written: ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ And ⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ with (The constant is called the electrical vacuum permittivity (units: Farad/m). Where F: Electrostatic force; k: Coulomb’s constant, q1, q2: Magnitude of the point charges, r: Distance of separation between the charges and is the unit vector carried by the support of MM′, oriented from M to M′, (in the direction from cause to effect). The force is repulsive if the charges have the same sign; it is attractive if they have opposite signs. According to the property of additivity of the electrostatic forces to which a charge q is subjected in the presence of two charges q1 and q2. The resultant of the forces is calculated as follows: 𝑞𝑞 𝑞𝑞2 𝐹𝑡𝑜𝑡 𝐹 +𝐹 𝑘 ⃗⃗⃗⃗ + 𝑘 𝑢 ⃗⃗⃗⃗ 𝑢 𝑟 𝑟 II.2. Electrical Field To define the electric field at a point in space, we place a small positive test charge q there and look at the Coulomb force F exerted on it, due to the presence of the surrounding electric charges that create the electric field. The electric field at this point is defined as the force per unit charge: ⃗ The electric field is therefore a vector quantity. The SI unit of electric field is the newton per coulomb (N.C-1). The test load must be small, so that it can be assumed that it does not itself Academic year : 2024-2025 3  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity disturb the surrounding electric field. At a distance from a point charge Q, the electric field is given by Coulomb's law: The principle of superposition that applies to Coulomb's law also applies to the electric field. To calculate the field created at a point by a set of n charges Q i , we first determine separately the field E1 due to Q1 , the field E2 due to Q2 , etc. The resulting field E is equal to the vector sum of the individual fields : ⃗ ⃗ +⃗ + +⃗ ∑⃗ II.3 Electric Potential (electric field potential, potential drop, the electrostatic potential) Electrical potential is in fact electrical potential energy per unit charge: where : VE = electrical potential (scalar) (J / C or V (volt) ), UE = electrical potential energy (scalar) (J: joule) and q = electrical charge (scalar) (C: coulomb). The electric potential V generated by a point charge Q decreases as a function of the distance separating the charge Q from the location P ( when The electrostatic potential energy, UE, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is: The potential at P due to the charge Q : III. Definition of a conductor in electrostatic equilibrium So far, we've only been interested in electrical charges and their effects. What happens to a conducting body in which charges are free to move? Academic year : 2024-2025 4  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity - In an insulator, charges remain where they were brought in (or removed). - In a conductor, the charges are mobile (or free) and are therefore liable to move under the action of even a very weak electric field. a- Definition of a conductor: A conductor is a material in which charges move when an electrostatic force is applied to them. In metals, only electrons are mobile. The network of positive charges has little mobility and can be considered fixed. In liquids and gases, ions are also mobile. b- Definition of a conductor in electrostatic equilibrium: the electrostatic equilibrium of a conductor is reached when no electric charge moves inside the conductor. This means that the distribution of charges remains constant over time. III.1. Properties of an equilibrium conductor 1- The electrostatic field inside the conductor is zero ⃗⃗⃗ because the charges do not move (the charge q is at rest) and 2- The potential inside the conductor is constant; it's an equipotential volume because the field is zero. The potential is constant, ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 3- All charges are placed on the surface of the conductor: If the conductor carries an electric charge, this charge is distributed over the outer surface of the conductor. There is no charge inside the conductor. The application of Gauss's theorem to a closed surface inside the conductor (The electric field flux E exiting a closed surface S is proportional to the total electric charge contained in the volume V bounded by this surface. The proportionality constant is 1 / ε 0 is the dielectric vacuum permittivity): ∯⃗ 4- The conductor surface is an equipotential region: The electric field ⃗ is perpendicular to the outer surface of the conductor. The field lines leave the conductor perpendicular to it, so: ∫ ⃗ ⃗⃗⃗ , because ⃗ is always perpendicular to the displacement vector ⃗⃗⃗ on the outer surface. III.2. Capacitance of a conductor in electrical equilibrium The charge Q of an isolated conductor (away from any other conductor) is proportional to its potential V. Consider a conductor at potential , a charge appears on its surface, defined by : Academic year : 2024-2025 5  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity ∯ If the potential becomes V1 , then V2 , then V3 , the charge becomes q1 , q2 , q3. Since charge- potential relationships are linear, we can write: The proportionality coefficient C, independent of q and V, is called the conductor's capacity. It is measured in farads (F), if q is in coulomb and V in volt. This constant is called the intrinsic capacitance of the insulated conductor: 1 Farad corresponds to a charge of 1 Coulomb at a potential of 1 Volt. However, sub-multiples of the Farad, are used. A conductor in electrostatic equilibrium carrying charge Q, let V be its potential and C its capacity. The Potential energy of a conductor in electrostatic equilibrium is written as: III.3. Capacitors III.3.1 Definitions A capacitor is a set of 2 conductors A and B in total influence. These two conductors are called the capacitor's armature. The capacitor's charge is that of its internal armature Q (Coulomb). VA is the potential of the internal armature and VB is the potential of the external armature, U= VA- VB is the potential difference of a capacitor (its unit is the volt). Its symbol in an electric circuit is: A B -Q Q The charge of a capacitor is written as: Q= C.U - The isolator (material placed between the armatures) increases the capacity of a capacitor. - The capacitance of a capacitor depends on the geometry of the armatures. - Capacity C is always positive. Examples  Planar capacitor: Consider two uncharged flat conductors. The distance between these two planes is d. The surface charge density is σ. Academic year : 2024-2025 6  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity This relationship remains approximately valid for two finite planes of area A surface area A and total charge Q, then :  Spherical capacitor  cylindrical capacitor Consider two coaxial conductive cylinders under total influence, one with charge +Q and the other with charge –Q Step 1: Calculating the E field The Gaussian surface is a cylinder of radius and height. Because of symmetry, the radial field is constant in the Gaussian surface. According to Gauss's Theorem: 𝑄 ∑ 𝐸 After calculation, we obtain: 2 2𝜋𝑟 𝜀 Step 2: Calculating the potential V The potential: ⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ∫ For so ∫ ∫ , Finally : ( ) 2 Step 3: Capacity calculation The charge is 2 ( ) III.3.2. Association of capacitors We can combine several capacitors of capacitance 2 to obtain a system with some effective capacitance C. The effective capacitance depends on the way the individual Academic year : 2024-2025 7  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity capacitors are combined. Two simple possibilities are : in series or parallel, or a combination of both: a- Capacitors in series: In a series connection of n capacitors, all capacitors store the same charge due to the total influence between the capacitor plates. D E On the other hand, the voltage between all the capacitors is equal to the sum of the voltages of the individual capacitors (series connection). + + + + Hence : + + For n capacitors in series: ∑ b- Capacitors in parallel In a parallel connection of n capacitors, all the capacitors have the same voltage U and the total charge is the sum of the charges of the individual capacitors. For two capacitors in parallel, we have: Figure (a) shows two capacitors arranged in parallel. In this case, the same potential difference is applied across both the capacitors. But the plate charges (±Q1) on capacitor 1 and the plate charges (±Q2) on the capacitor 2 are not necessarily the same: 𝑄 𝐶 𝑉 𝑄 𝐶 𝑉 , The equivalent capacitor is one with charge: 𝑄 𝑄 + 𝑄 and potential difference V: 𝑄 𝐶 𝑉 𝐶 𝑉+𝐶 𝑉 The effective capacitance C is : 𝑪 𝑪𝟏 + 𝑪𝟐 Parallel combination of (a) two capacitors, (b) n capacitors The general formula for effective capacitance for parallel combination of capacitors, Fig. (b), + + + follows similarly, + + + + IV. Energy stored in a capacitor Academic year : 2024-2025 8  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity IV.1. Electrostatic energy A capacitor can be charged by connecting a generator between its plates. This generator transfers charges from one armature to the other. The result is an increase in the capacitor's electrostatic potential energy. To calculate this energy, we assume that is the charge of the capacitor at a certain instant during charging. At this instant, the potential difference between the two armatures is. The work required to transfer an infinitesimal charge from the negative to the positive armature , when the charge of the armature (A) goes from the value to the very close value + is given by: Total work W, stored in the capacitor, when the charge on the armature (A) changes from zero (capacitor discharged) to a value , is obtained by summing the elementary variations : ∫ 2 This work is stored in the form of electrical potential energy. Since we have the relationship , where is the potential difference, this electrical potential energy can be expressed as a function of the potential difference between the armatures : 2 2 IV.2. Energy density The electrostatic energy of a capacitor can be considered to be stored by the electric field ⃗⃗⃗ in the volume it occupies in space. To do this, we introduce the energy density of the electric field ⃗ per unit volume: The electrostatic energy stored by the electric field will be: ∫ This formula is always applicable regardless of the electric field. IV.3. Capacities with dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. A dielectric is a non-conductive material such as glass, rubber, etc. When a dielectric is inserted in all the space between the plates of a capacitor, the capacitance of the capacitor increases, multiplied by a factor called the dielectric constant of the material: Where is the capacity of the capacitor without the dielectric. κ varies from one material to another, for example Academic year : 2024-2025 9  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Material Dielectric constant Vacuum 1 Air 1.00059 3 Bakelite 4.9 24 Mylar 3.2 7 Nylon 3.4 14 Porcelain 6 12 Pyrex glass 5.6 14 Paper 3.7 16 The table also gives the values of the maximum electric field that can be supported by the dielectric. V. Electric current and current density V.1. Conductors and Insulator An electric current is an ordered movement of charged particles. This uniform motion of electrons is what we call electricity or electric current. A conductor is anybody, whether solid, liquid or, in some cases, a gas, that possesses mobile charged particles. These are electrons, but also positive or negative ions, i.e. atoms or molecules that have lost or captured one or more electrons.  Current in a metallic conductor Conductors are materials in which current flows easily, because free electrons are plentiful and can move easily from one place to another. Most metals are good conductors, with silver being the best conductor of all, followed by copper. Copper is generally used as a conductor in electrical wires. In a metallic conductor, an electric current is a displacement of electrons. Charges +Q and –Q put at the ends of a metallic cylinder. The electrons will drift because of the electric field created to neutralise the charges. The current thus will stop after a while unless the charges +Q and –Q are continuously replenished. electron  Current in a liquid In liquids, the electric current is made up of negative ions and positive ions always moving in opposite directions.  Insulating materials are poor conductors, with few free electrons. They're most often used to prevent current flow (glass, plastic). Academic year : 2024-2025 10  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity  Semiconductors tend to be insulators, but become conductors if the temperature is raised or if they contain impurities. Semiconductors are the basis of electronic circuits. V.2. Intensity of Electric Current Electric current is the ordered movement of electric charges under the effect of an electric force (obtained by the application of a potential difference, which gives rise to an electric field). This movement can be of different natures: o Movement of electrons in a conductor (metals) o Movement of ions in a liquid (electrolysis, human body, batteries) o Movement of positive charges (holes) in semi-conductors Electrons, ions and holes are charge carriers. The intensity of an electric current, noted , represents the quantity of electric charge (expressed in coulomb) that has passed through a section of conductor over a period of. The international unit of electric current is the Ampere, noted [A]. The conventional direction of current, also known as conventional current, is arbitrarily defined as the direction in which positive charges flow. The electrons, the charge carriers in an electrical circuit, flow in the opposite direction of the conventional electric current. There are essentially two types of current.  Direct current: Direct current (DC) is an electric current that is uni-directional, so the flow of charge is always in the same direction. As opposed to alternating current, the direction and amperage of direct currents do not change. It is used in many household electronics and in all devices that use batteries. The intensity of DC is constant over time:  Alternative Current: In alternating current (AC), the movement of electric charge periodically reverses direction. The variable intensity of AC is identical to itself at regular intervals period + + 2 + V.3. Electric current density For the moment, we can limit ourselves to a single type of carrier, electrons for example. Under the action of an electric field , each electron acquires a velocity , the average velocity of all electrons (also known as the entrainment velocity), and the volume charge of the medium, we define the electric current density vector at any point in the medium as : Academic year : 2024-2025 11  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity and its unit is the Ampere per square meter ( ). Since where is the number of electrons per unit volume volume and is the absolute value of the electron's charge: 𝑗 𝑛𝑒𝑣 Current density vector The current flowing in the wire is related to the density by: ∫ The current in a circuit is the flow through the cross-section of the wire of the current density. The direction of the current is given by the direction of the current density vector. VI. OHM’s LAW A basic law regarding flow of currents was discovered by G.S. Ohm in 1828. Imagine a conductor through which a current is flowing and let be the potential difference between the ends of the conductor. Then Ohm’s law states that: (The electric current through a conductor between two points is directly proportional to the voltage across the two points) So (where the constant of proportionality R is called the resistance of the conductor). The SI unit of resistance is Ohm, and is denoted by the symbol. The resistance R not only depends on the material of the conductor, but also on the dimensions of the conductor. At a given temperature, the resistance of a conductor with a uniform cross-section is written as (POUILLET law): Where, the constant of proportionality depends on the material of the conductor but not on its dimensions. is called resistivity. the length (meter) and S: the cross sectional area. In an electrical conductor, the current density is proportional to the local electrostatic field. This relationship can be expressed as follows: ⃗ The proportionality coefficient γ is called the conductivity of the medium. Conductivity, , is the inverse of resistivity : Academic year : 2024-2025 12  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Table of Resistivity and Conductivity at 20°C of some materials  The resistance of a conductor varies with temperature (Mathiessen law). The resistivity of a material is found to be dependent on the temperature. The resistivity of a metallic conductor is approximately given by: ( + ) Where is the resistivity at temperature T, is the same at a reference temperature T0, is called the temperature coefficient of resistivity at 20°C (e.g. copper 0.0038). For metals, is positive The following is a summary of the various rules of Ohm's law: Ohms Law matrix Table The potential difference Potential difference (p.d.d.) or electric voltage is the difference in positive (+) and negative (-) charges between the 2 terminals of a generator. This value is expressed in VOLT (symbol V) and can be measured with a voltmeter. Academic year : 2024-2025 13  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Current and voltage orientation convention: Receiver convention: Generator convention: VII. Joule’s Law (Joule effect formula) In addition to current, voltage and resistance, a fourth parameter is very important in electricity: Electrical power. Electrical power: Power is a measure of the amount of work that can be delivered in a given time. Power is symbolized by the letter P and its unit of measurement is the watt (W). We can therefore conclude that electrical power is directly proportional to voltage and current: power = voltage x current. Hence : Power dissipation through a resistor takes the form of heat. If we replace I in the power formula by its equivalent in Ohm's law, we obtain : P: is the power (energy per unit time) converted from electrical energy to thermal energy, The joule’s first law shows the relationship between heat produced by flowing electric current through a conductor in direct current: W: energy dissipated in heat, t: denote time (Second s) VIII. KIRCHHOFF’S RULES Electric circuits generally consist of a number of resistors and cells interconnected sometimes in a complicated way. Kirchhoff's laws are physical properties that apply to electrical circuits. They are named after the German physicist Gustav Kirchhoff, who established them in 1845. Two rules, called Kirchhoff’s rules are very useful for analysis of electric circuits:  Junction rule  Loop rule Definitions: - A branch is a set of dipoles connected together and carrying the same current. - A node is the junction point of at least three conductors. - A loop is a closed path made up of successive network branches. Academic year : 2024-2025 14  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity - An electrical dipole is an electrical component with two terminals (lamps, switches, generators, batteries, diodes, resistors, motors, etc.). ). A linear dipole is an electrical dipole whose current i(t) through it and voltage v(t) at its terminals are linked by a linear operator. VIII.1. Junction rule (Kirchhoff’s first Rule) It states that the sum of all currents directed towards a junction (point A) in an electrical network is equal to the sum of all the currents directed away from the junction (see figure below). In other words, the algebraic sum of all currents at a junction is zero. Kirchhoff’s first rule tells us that there is no accumulation of charge at any point if steady current flows in it. 𝑛 At each junction of a circuit, we have: ∑ 𝛼𝑘 𝐼𝑘 𝑘 Example: where 𝛼𝑘 = +1 when the current is incoming and 𝛼𝑘 = - 1 when it is outgoing. 1 + 2 + 3 − 4 = 0 or 4 = 1 + 2 + 3 VIII.2. Loop Rule (Kirchhoff’s Second Rule) This rule is an application of law of conservation of energy for electrical circuits. This law states that “in a loop of an electrical network, the sum of the voltages along this loop is always zero”. In other words, if we go around a loop and add up all its voltages (paying attention to the direction), the sum will be zero. Following the red direction, the voltages can be listed as follows: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero IX- Resistance associations The equivalent resistance of a combination of resistors depends on both their individual values and how they are connected. The simplest combinations of resistors are series and parallel connections (see figure below). Academic year : 2024-2025 15  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity - In a series circuit, the output current of the first resistor flows into the input of the second resistor; therefore, the current is the same in each resistor. - In a parallel circuit, all of the resistor leads on one side of the resistors are connected together and all the leads on the other side are connected together. In the case of a parallel configuration, each resistor has the same potential drop across it, and the currents through each resistor may be different, depending on the resistor. The sum of the individual currents equals the current that flows into the parallel connections. 1- Resistors in Series (Series Association) Consider Figure below, which shows three resistors in series with an applied voltage equal to Vab. Since there is only one path for the charges to flow through, the current is the same through each resistor. The equivalent resistance of a set of resistors in a series connection is equal to the algebraic sum of the individual resistances. The current through the circuit depends on the voltage supplied by the voltage source and the resistance of the resistors. For each resistor, a potential drop occurs that is equal to the loss of electric potential energy as a current travels through each resistor. According to Ohm’s law, the potential drop V across a resistor when a current flows through it is calculated using the equation V=R.I , where I is the current in amps (A) and R is the resistance in ohms (Ω). Since energy is conserved, and the voltageis equal to the potential energy per charge, the sum of the voltage applied to the circuit by the source and the potential drops across the individual resistors around a loop should be equal to zero: 2 + 2 + + + Solving for : + + Academic year : 2024-2025 16  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity So the equivalent resistance is just the sum of the resistances of the individual resistors. Any number of resistors can be connected in series. If N resistors are connected in series, the equivalent resistance is: + 2 + + ∑ 2- Resistors in parallel : Parallel association Each portion of the circuit shown in the figure below is supplied with an electric current which is distributed between the resistors (see figure), such that: The equivalent resistance is given by : ∑ 3- Electric circuits An electric circuit is a set of electrical components such as resistors, capacitors, diodes...etc. The electrical conductors are carried by an electric current. Electric Circuit is the closed loops or paths, in which the current flows. Therefore, the electrokinetics of an electric circuit consists in finding the current intensity and voltage for each point in the circuit.  Direct Current Circuit or DC Circuit is a closed electrical circuit in which the flow of electricity is in one direction. DC Circuit has a DC Power Supply which produces Direct Current in the circuit. As opposed to alternating current, Direct Current has a fixed magnitude and flows in one direction only. Direct Current Circuit forms a major backbone of the electronics industry.  Alternative Current Circuit or AC circuits are powered by an alternating source such as alternating currents or voltages which are sinusoidal and change periodically in direction and magnitude. In other words, voltage or current oscillates in a sine wave pattern and varies with time. 3.1. Generators An electric circuit needs a power source to supply it with energy. It is therefore necessary to connect these circuits with a device called a generator (electromotive forces) to transport electrical charges. Academic year : 2024-2025 17  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity There are two categories of electric generators: a- Voltage source (Voltage generator) In the case of a voltage generator, the electromotive force (EMF) is equal to the potential difference between its terminals when no current flows. The internal resistance r of an electric generator is responsible for the drop in voltage supplied by the generator as the current it delivers increases. 𝑈 𝑒 𝑟𝐼 b- Current source (Current generator) A current generator is a dipole characterized by a constant current, regardless of the potential difference between its terminals. 3.2. Wheatstone Bridge You have learnt that a resistance can be measured by Ohm’s law using a voltmeter and an ammeter in an electrical circuit. But this measurement may not be accurate for low resistances. To overcome this difficulty, we use a wheatstone bridge. It is an arrangement of four resistances which can be used to measure one of them in terms of the other three. As an application of Kirchhoff’s rules consider the circuit shown in figure below, which is called the Wheatstone bridge. The bridge has four resistors R1, R2, R3 and R4. Across one pair of diagonally opposite points (A and C in the figure) a source is connected. The Kirchhoff’s junction rule applied to junctions D and B (generally 𝐼𝐺 ). Immediately gives us the relations 𝐼 𝐼 𝑎𝑛𝑑 𝐼 𝐼 Next, we apply Kirchhoff’s loop rule to closed loops ADBA and CBDC. The first loop gives: –𝑅 𝐼 + + 𝐼 𝑅 and the second loop gives, upon using 𝐼 𝐼 𝑎𝑛𝑑 𝐼 𝐼 –𝑅 𝐼 + + 𝐼 𝑅 2 𝐼 𝑅2 From Eq. 1, we obtain, 𝐼2 𝑅 𝐼 𝑅 From Eq. 2, we obtain, 𝐼2 𝑅 Hence, we obtain the condition: 𝑅2 𝑅 𝑅 𝑅 This last equation relating the four resistors is called the balance condition for the galvanometer to give zero or null deflection. The bridge then is balanced, and from the balance condition the value of the unknown resistance is given by, Academic year : 2024-2025 18  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity A practical device using this principle is called the meter bridge. IX. Capacitor charging and discharging Consider an RC dipole consisting of a resistor R and a capacitor (condenser) of capacitance C connected to an e.m.f. generator e. Initially, the capacitor is not charged: q0 =0. We are interested in determining the charge q of the capacitor at any instant. 1- Expression of capacitor charge  At time t, according to the Loop law, we have: 𝑖 𝑈𝑅 + 𝑈𝐶 𝐸  the voltage across the resistor is: 𝑑𝑞 𝑈𝑅 𝑅 𝑖 𝑡 𝑅 2 𝑑𝑡  the ddp (voltage) across a capacitor of capacitance 𝐶 𝑞 𝑡 𝑈𝐶 We obtain: 𝐶 + Equation (3) is a 1st-order differential equation. It can be solved either:  By the separation of variables method.  By solving the general solution The integration constants are found by using boundary conditions.  A condition at t = 0.  A condition when tends to infinity.  The resolution of (3) by the separation of variables method Where is a time-dimensional quantity, known as the circuit's characteristic time (s). The constant A is determined from the initial conditions. 𝑡 𝑞 𝑡 𝐶 𝐸( 𝑒 𝑅𝐶 ) Academic year : 2024-2025 19  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity Current and voltage across the capacitor. The current intensity during capacitor charging is given below: The voltage across the capacitor is : ( ) Graphical representation  Capacitor energy Initially, the capacitor's energy is zero, since its charge is zero. When it is charged, the voltage between its terminals is E (generator e.m.f.) and its charge is and its energy is given by: 2 2 The energy supplied by the generator is : In charging the capacitor of an RC circuit, half of the energy drawn from the battery is stored in the capacitor while the other half is dissipated as heat by the resistor (joule effect): 2- Capacitor discharge Initially, the capacitor is fully charged. Its initial charge is given by. It discharges into the resistor. In the closed discharge circuit, the sum of the voltages is zero. In this case, we obtain : Academic year : 2024-2025 20  Electricity course National Higher School of Cybersecurity Basic training in Cybersecurity Department of Basic Training in Cybersecurity The equation is a 1st-order differential equation with no second member, with attention to the condition. Applying the method of variable separation, we obtain the following differential equation: + + Considering the following initial condition:. Consequently: Current and voltage across the capacitor The current during capacitor discharge is: The voltage across the capacitor is: Charge Voltage Current Academic year : 2024-2025 21