Topic 3: Electric Circuits

1. Moving charge and electric current:

• Moving charges are a fundamental concept in physics, especially in the study of electromagnetism.
• A moving charge is a charge that is in motion, like a proton or electron moving through space.
• When a charge moves, it creates a magnetic field around itself, which can interact with other charges and magnetic fields.
  – They create a magnetic field, which can be calculated using the Biot-Savart law or the Lorentz force equation.
  – The magnetic field generated by a moving charge depends on its velocity, direction, and the charge’s magnitude.
  – Moving charges can experience a force due to the magnetic field they generate, known as the self-force.
  – The concept of moving charges is crucial in understanding various phenomena, such as electric currents, electromagnetic induction, and particle accelerators.
• Electric current is the flow of electric charge, typically in the form of electrons moving through a conductor, such as a wire.
• [math] \text{Electric current } I \, (\text{A}) = \frac{\text{amount of charge flowing, } \Delta Q \, (\text{C})}{\text{time to flow, } \Delta t \, (\text{s})} \\ I \, (\text{A}) = \frac{\Delta Q \, (\text{C})}{\Delta t \, (\text{s})} [/math]
• – Electric current is the rate of flow of electric charge, typically measured in SI unit amperes (A).
  – It’s the movement of electrons from a negative terminal to a positive terminal, driven by a potential difference (voltage).
• There are two types of electric current:
  – Direct Current (DC): Flows in one direction only, like in a battery or electronic device.
  – Alternating Current (AC): Periodically reverses direction, like in a household power supply.
  – Resistance, impedance, and capacitance can affect electric current in a circuit.
• Many microelectronic circuits, like the printed circuit boards inside many computer systems, function with currents of the order of microamps ([math] μ \text{A} , 10^{-6} \text{A} [/math]), whereas electronic circuits, like those that control home appliances, often operate with considerably lower currents, milliamps ([math] \text{mA} , 10^{-3} \text{A} [/math]).

  
  Figure 1 Electric currents can also
  be due to the flow of ions

• An electrolyte is a substance that helps conduct electricity in a solution, such as a liquid or a gel.
• Electrolytes are essential for various biological and chemical processes.
• Electrolytes are substances that dissolve in water or other polar solvents to produce ions, which can conduct electricity.
• Positive ions (cations) move toward the negative electrode
• Negative ions (anions) move towards the positive electrode.
• Electrons flow in the external circuit from the positive electrode to the negative electrode.

⇒ Example

If a car battery delivers 450 A for 2.5 seconds, calculate the total charge flowing, ∆Q.

• Given data:
  Time = t = 2.5 s
  Current = I =   450 A
• Find data:
  Flow of charges = ∆Q =  ?
• Formula:
• [math] \Delta Q = I * \Delta t [/math]
• Solution:
• [math] \Delta Q = I * \Delta t [/math]
• Put values
• [math] \Delta Q =450 * 2.5 \\ \Delta Q = 1125C[/math]

2. Potential difference and electromotive force:

• Only the speed at which charged particles—typically electrons—move through a circuit is measured by electric current.
• It provides no information on the electrical energy used in circuits.
• Potential difference (pd), represented by the symbol V and expressed in volts (V), is the SI unit used to represent the electrical energy in circuits.
• A voltmeter is placed across and in parallel with an electrical component to detect the potential difference across it.
• Potential difference, also known as voltage, is the difference in electric potential energy between two points in a circuit.
• It’s the driving force behind electric current.
• This electrical energy is transferred into heat, light and other more useful forms of energy by the components.
• [math] \text{Potential difference } V \, (\text{V}) = \frac{\text{Electrical work done by the charge } W \, (\text{J})}{\text{Charge flow } Q \, (\text{C})} \\ V \, (\text{V}) = \frac{\text{W} \, (\text{J})}{\text{Q} \, (\text{C})} \\ 1 \, \text{V} = 1 \, \text{J} \, \text{C}^{-1} [/math]
• On the other hand, the energy changes associated with power supplies like batteries, generators, and main power supply units cannot be well described by potential difference.
• These apparatuses convert many types of energy, such chemical energy, into electrical energy.
  – Measured in volts (V)
  – The energy per unit charge that a source provides to drive current.
  – Can be thought of as the “pressure” that pushes electric charge through a circuit.
  – Can be generated by various sources, such as:

   

  • Batteries (chemical energy converted to electrical energy)
  • Generators (mechanical energy converted to electrical energy)
  • Solar cells (light energy converted to electrical energy)
• The law of conservation of energy can now be written in terms of emf and pd. In a series circuit, where the components are connected one after another in a complete loop.
• The total electrical energy per coulomb transferring into the circuit (the sum of the emfs in the circuit) must equal the energy per coulomb transferring into other forms of energy (the sum of the pds).
• 
  Figure 2 A circuit diagram with attach a battery provide potential difference
• Potential difference amount by volt meter.
• Electrical current measured by ampere mater.

3. Resistance:

• Resistance (R) is the opposition to the flow of electric current through a conductor. Here are some key points about resistance.

  
  Figure 3 Electric charges flow into
  a circuit with attach a resistance (R)

  – Measured in ohms (Ω).
  – Depends on the material, length, and cross-sectional area of the conductor.
  – Increases with increasing length and decreasing cross-sectional area.
  – Decreases with increasing temperature (in most cases).

• The substance of the circuit obstructs the passage of the charge when current passes through it, such as through the metal connecting wires.
• The oscillating positive ion cores of the metal structure are collided with by the electrons as they move through the metal on a tiny level.
• The electrons’ electrical energy is transferred to the metal’s structure via the collisions with the positive ion cores, which increases the vibration of the metal ion cores and heats the wire.
• As the temperature rises, so does the resistance.
• As the temperature rises, the positively charged, vibrating ion cores move around more and obstruct the passage of electrons.
• This is in opposition to the electron gas’s passage through the structure.
• Electrical insulators are parts that allow very little electricity to flow through them due to their high resistance.
• Ohm’s Law states that the current flowing through a conductor is directly proportional to the voltage applied across it, and inversely proportional to the resistance of the conductor.
• Mathematically, it’s expressed as:
• [math] I ∝ V \\ V = IR [/math]
• When applied to metal wires at constant temperature, it is often a specific situation.
• However, by utilizing the idea of resistance, a practical, well-known mathematical equation may be created that defines the ohm (Ω), the unit of resistance, using the connection between I and V.
• [math] R = \frac{V}{I} [/math]
• Put differently, a component must have a resistance of 1Ω if a potential difference of 1V results in a current of 1A flowing through it.

4. Charge conservation:

• Charge conservation states that the total electric charge in a closed system remains constant over time. In a circuit, this means that the total amount of electric charge (electrons) is conserved.
• Here’s how this leads to the distribution of current:
  1. Kirchhoff’s Current Law (KCL): The sum of currents entering a node (junction) is equal to the sum of currents leaving the node. This is a direct consequence of charge conservation.
  2. Conservation of charge: The total charge in the circuit is constant. Therefore, the rate at which charge flows into a component (current) must equal the rate at which charge flows out.
  3. Current distribution: As a result, the current in a circuit distributes itself in a way that ensures the conservation of charge. This means that the current will split or combine at nodes to maintain the overall balance of charge.
• Think of it like water flowing through pipes: the amount of water (charge) entering a junction must equal the amount leaving. This ensures the water (charge) is conserved, and the flow (current) is distributed accordingly.

⇒Kirchoff’s First Circuit Law – the law of current:

• “At a circuit junction, the sum of the currents flowing into the junction equals the sum of the currents flowing out of the junction”.

  
  Figure 4 A circuit junction

• Figure 4 shows a circuit junction with two currents ([math] I_1 \; \text{and} \; I_2 [/math]) flowing into the junction and three currents flowing out of the junction ([math] I_3, \; I_4, \; \text{and} \; I_5 [/math]).
• Kirchoff’s First Circuit Law states:
• [math] I_1 + I_2 = I_3 + I_4 + I_5 [/math]
• Kirchoff’s First Circuit Law is demonstrated in an actual circuit using ammeters in Figure 5
• Since conventional current flows from positive to negative, ammeter [math] A_1 [/math] measures the current entering the junction shown on the diagram.
  At a junction, current divides or recombines, therefore [math] A_2 [/math] and [math] A_3 [/math] measure the current leaving the junction. According to Kirchoff’s First Circuit Law, you can:
• [math] A_1 = A_2 + A_3 [/math]
• Written more generally in mathematical notation the law can be summarized by:
• [math] \sum I_{\text{out  of  junction}} = \sum I_{\text{out  of  junction}} [/math]
• or, in other words, current is conserved at junctions.
• 
  Figure 5 Circuit diagram showing Kirchoff’s First Circuit Law.
• Looked at from a slightly different perspective, as current is the rate of flow of charge, or
• [math] I = \frac{\Delta Q}{\Delta t}[/math]
• “At a circuit junction, the sum of the charge flowing into the junction equals the sum of the charge flowing out of the junction (per second)”.
• [math] \sum Q_{\text{into junction}} = \sum Q_{\text{out of junction}} [/math]

⇒ Kirchoff’s Second Circuit Law – the law of voltages:

• “In a closed-circuit loop, the sum of the potential differences is equal to the sum of the electromotive forces”.
• Figure 6 shows a single closed loop series circuit. In this case, there is one emf and two pds, and Kirchoff’s Second Circuit Law says that:
• 
  Figure 6 Circuit diagram showing Kirchoff’s Second Circuit Law.
• [math] \varepsilon = V_1 + V_2 [/math]
• Or more generally, using mathematical notation, for any closed-circuit loop:
• [math] \sum \varepsilon = \sum V [/math]
• If the circuit is extended to make it a parallel circuit such as Figure 7:
• This parallel circuit is effectively made up of two series circuits: ABCD and AEFD, so:
• [math] \varepsilon = V_1 + V_2 [/math]
• and
• [math] \varepsilon = V_3 + V_4 [/math]
• 
  Figure 7 Circuit diagram showing Kirchoff’s Second Circuit Law in a parallel circuit
• Or
• [math] V_1 + V_2 =V_3 + V_4[/math]
• Kirchoff’s Second Circuit Law holds true for all circuits; but, when it comes to parallel circuits, the circuit has to be seen as a collection of separate series circuits sharing a single power source.
• This equation is founded on the principle of energy conservation: when the charge moves through the circuit components, the energy per coulomb that the battery provided to it, or ε, is converted by the charge into different types of energy.

5. Resistor networks:

• Every component provides a specific amount of resistance to the circuit when it is joined to other components to create functional circuits.
• Whether a component is linked in series or parallel with other components determines how the additional resistance affects the circuit.
• The total resistance of components linked in parallel or series may be determined using a few straightforward formulas.

⇒ Resistors connected in series:

• Consider the resistor network shown in Figure 8 and 9.

  
  Figure 8 Circuit diagram showing
  resistor combinations in series.
  
  Figure 9 Circuit diagram showing
  equivalent resistor

• Figure 9 represents the single resistor that could replace the three resistors in series in Figure 8.
• Using Kirchoff’s Circuit laws and Ohm’s law leads to:
• [math] \varepsilon = V_1 + V_2 + V_3 [/math]
• and
• [math] \varepsilon =V_T [/math]
• where
• [math] V_T = V_1 + V_2 + V_3  \qquad (5) [/math]
• So,
• [math] V = IR [/math]
• Put in equation according to resistor and volt then
• [math] IR_T = IR_1 + IR_2 + IR_3 \\ IR_T = I(R_1 + R_2 + R_3) [/math]
• So,
• [math] R_T = R_1 + R_2 + R_3 [/math]
• and for a series network of n resistors:
• [math] R_T = R_1 + R_2 + R_3 + \dots + R_n [/math]
• or, using sigma notation
• [math] R_T = \sum_{i=1}^{n} R_i [/math]

⇒ Resistors connected in parallel:

• Consider the following circuit:
• 
  Figure 10 Circuit diagram showing resistors connected in parallel
• In the right-hand circuit one resistor, RT, has been used to replace all three resistors arranged in parallel in the left-hand circuit.
• Again, using Kirchoff’s Circuit laws and Again, using Kirchoff’s Circuit laws and the definition of resistance,
• [math] V = IR [/math]
• Kirchoff’s First Circuit law says:
• [math] I_T = I_1 + I_2 + I_3 [/math]
• And
• [math] I = \frac{V}{R} [/math]
• So as the potential difference, , is the same across all of the resistors:
• [math] I_T = \frac{V_T}{R_1} + \frac{V_T}{R_2} + \frac{V_T}{R_3} \\ I_T = V_T \left( \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \right) [/math]
• Rearranging
• [math] \frac{I_T}{V_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \\ \frac{I_T}{V_T} = \frac{1}{R_T} [/math]
• So,
• [math] \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} [/math]
• For a network of n resistors connected in parallel
• [math]  \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots + \frac{1}{R_n} [/math]
• or using sigma notation:
• [math]  \frac{1}{R_T} = \sum_{i=1}^{n} \frac{1}{R_i}[/math]
• To summarize, the total resistance of a series of resistors is equal to the sum of its individual resistances.
• The reciprocal of the total resistance for resistors connected in parallel is equal to the reciprocal of each individual resistance.
• The resistance of a parallel arrangement of resistors is always smaller than the resistance of any individual resistor in that combination.

6. Electrical power in circuits:

• An incredibly intricate work of electronic engineering is the integrated circuit of an iPhone.
• It entails the communication between thousands of tiny electrical components.
• These are made to cooperate in order to regulate the phone’s numerous features.
• Despite their complexity, all of the circuits are founded on a few fundamental concepts and laws of circuitry.
• Integrated circuits are excellent at single-step, basic tasks.
• They merely complete them in predetermined order and extremely swiftly.
• The total power that each individual electrical component in a circuit transfer equals the total power that the circuit transfers.
• A component’s ability to transfer electrical power is determined by the potential difference across it and the current that passes through it. Power is often defined as:
• power = rate of doing work or rate of energy transfer
• or
• [math] P = \frac{\Delta W}{\Delta t}[/math]
• Thus, the rate at which electrical work is completed may be used to describe electrical power in an electrical context.
• A component’s resistance, current, and potential difference may all be utilized to compute the electrical power it transfers.
• This power can be calculated using a variety of formulas.
• The fundamental equations for electric current, potential difference, and resistance definition are rearranged to create these equations.
• [math] \text{Current} = \frac{\text{Charge}}{\text{Time}} [/math]
• In terms of symbols:
• [math] I = \frac{Q}{t} \\ Q = I t \quad \text{(1)} \\ \text{Potential difference} = \frac{\text{Electrical energy}}{\text{Charge}} [/math]
• In terms of symbols:
• [math] V= \frac{W}{Q} \\ Q= \frac{W}{V}[/math]
• Substitute in equation 1
• [math] It =\frac{W}{V} [/math]
• Then
• [math]IV = \frac{W}{t} [/math]
• So,
• [math] P = IV \qquad  (2) [/math]
• By using the Ohm’s law
• [math] V = IR [/math]
• Put in equation 2
• [math] P= I(IR) [/math]
• Then,
• [math] P = I^2 R \qquad (3) [/math]
• If we rearranging the ohm’s equation then we get
  [math] I= \frac{V}{R} [/math]
• Put in equation 3
• [math] P = \left(\frac{V}{R}\right)^2 R \\ P = \frac{V^2}{R^2} \cdot R [/math]
• So,
• [math] P = \frac{V^2}{R}  \qquad (4) [/math]
• In the form of energy
• [math] \text{Power} = \frac{\text{Energy}}{\text{Time}} \\ \text{Energy} = V \cdot I \cdot t [/math]
• The connection P = IV is very helpful because it makes it possible to calculate electrical power using current and potential difference, two quantities that are simple to measure and keep track of.
• Real-time monitoring of a circuit’s electrical power consumption is made possible by data-logging ammeters and voltmeters.
• This enables battery-powered devices, like laptops, tablets, and smartphones, to show their remaining energy and estimate how long they will last before needing to be charged.
• A resistor is a passive electrical component that reduces the voltage or current in a circuit.
• Symbol:
• The symbol for a resistor is a zigzag line or a rectangular box with a “R” inside.
• Units:
• Resistors are measured in Ohms (Ω), with common values ranging from a few ohms to millions of ohms.

7. Current-potential difference (I-V) graphs:

• Current-potential difference (I-V) graphs are a powerful tool for understanding the behavior of electrical components. Here’s a brief guide on how to sketch, recognize, and interpret I-V graphs for ohmic conductors, filament bulbs, thermistors and diodes.
• Ohmic Conductors (e.g., Resistors):
  – Sketch: A straight line with a positive slope
  – Recognize: The graph passes through the origin (0,0) and has a constant slope
  – Interpret:
• – The slope represents the resistance (R)
• – The x-intercept represents the potential difference (V)
• – The y-intercept represents the current (I)
• – The graph shows a linear relationship between I and V
• As current passes through a conventional tungsten filament lamp, electrical energy is converted into both light and heat.
• More kinetic energy is transferred when the current rises because there are more electron collisions with the tungsten lattice’s positive ion centers.
• The resistance rises as a result of the positive ion cores’ increased amplitude vibrations.
• A greater temperature is caused by a higher current, and a higher resistance follows from that.
• Figure 11 illustrates a filament lamp’s electrical characteristics.
• A non-ohmic component is a component that does not obey Ohm’s law; i.e. current is not proportional to the potential difference applied across it.
• No matter which way the current flows through them, components like filament bulbs and fixed resistors have the same properties.
• Both the forward and reverse bias forms on their V-I graphs are the same.
• This is not how components like semiconductor diodes
• Typically used in electrical circuits, diodes function as one-way gates to stop current from returning to the source.
• Because they may be utilized in circuits to convert alternating current (ac) into direct current (dc), they are very helpful in major power supply.
• 
  Figure 12 A semiconductor diode – symbol and picture. The current will only flow in the direction of the arrow (anode to cathode).
• Diodes only conduct in forward bias, or in the direction indicated by the arrow on the symbol.

  
  Figure 13 Electrical characteristic of a semiconductor diode

• The component itself typically has a distinct colored ring at the forward bias end to indicate this orientation.
• Since diodes don’t conduct under reverse bias, their resistance is infinite in this situation.
• In forward bias, diodes have extremely low resistance.
• Typically, diodes’ electrical properties are shown as current (y-axis)–potential difference (x-axis) diagrams.

⇒ Thermistors:

• Thermistors are temperature-sensing devices that exhibit a change in electrical resistance when their temperature changes.
• They’re made from specialized materials that are designed to detect temperature changes.
• 
  Figure 14 Thermistor circuit symbol, resistance – temperature graph and shape
• Types of Thermistors:
  – NTC (Negative Temperature Coefficient) Thermistors: Resistance decreases as temperature increases
  – PTC (Positive Temperature Coefficient) Thermistors: Resistance increases as temperature increases
• Characteristics:
  – High sensitivity and accuracy
  – Fast response time
  – Low power consumption
  – Small size and low cost
• Applications:
  – Temperature measurement and control
  – Thermal protection and monitoring
  – Automotive and industrial applications
  – Medical devices and equipment
  – Consumer electronics and appliances
• At a low temperature of 20°C, the resistance of the thermistor is high and so the current (and the V/R ratio) is low.
• [math] R= \frac{V}{I} \\
  R = \frac{6 \, \text{V}}{0.002 \, \text{A}} \\
  R = 3000 \, \Omega [/math]
• A higher temperature of 80°C, the resistance is lower and the V/R ratio is higher.
• [math] R= \frac{V}{I} \\
  R = \frac{11 \, \text{V}}{0.010 \, \text{A}} \\
  R = 1100 \, \Omega [/math]
• The current flowing through the thermistor is therefore high.

8. Resistivity:

• Resistivity is a fundamental property of materials that describes their ability to resist the flow of electric current.
• It’s defined as the ratio of the electric field strength to the current density, and it’s typically denoted by the symbol ρ (rho).
• Resistivity is measured in units of ohm-meters (Ω m), and it’s a characteristic property of each material.

  
  Figure 15 Resistance and resistivity

• Some materials have high resistivity, meaning they resist current flow, while others have low resistivity, meaning they conduct current easily.
• Here are some key points about resistivity:
  – Resistivity is a tensor quantity, meaning it can have different values in different directions in an anisotropic material.
  – Resistivity is temperature-dependent, typically increasing with temperature.
  – Resistivity is affected by impurities, defects, and other material properties.
• As the conductor’s length (l) rises, more positive ion cores obstruct the flow of electrons through the conductor, raising the resistance (R). Actually, the resistance (R) doubles if the length (l), doubles. Thus, it follows that
• [math] R \propto I \qquad (1) [/math]
• The resistance of the conductor decreases with increasing cross-sectional area, A.
• In this case, if the cross-sectional area, A, doubles.
• Then the resistance, R, halves. This means that
• [math] R \propto \frac{1}{A} \qquad (2) [/math]
• Combining both of these proportionality statements together:
• [math] R \propto \frac{1}{A}  [/math]
• and replacing the proportionality sign and adding a constant of proportionality, ρ, we have:
• [math] R = \text{const} \frac{l}{A} \\ R = \frac{\rho * l}{A} [/math]
• Where ρ (‘rho’) is called the electrical resistivity of the material.
• Resistivity is the property that gives the intrinsic resistance of the material independent of its physical dimensions, such as length and cross-sectional area.
• Resistivity has the units of ohm meters, Ωm, and is defined by the rearranged form of the equation:
• [math] \rho = \frac{R A}{l} [/math]
• where R is the resistance (measured in ohms, Ω).
• A is the cross-sectional area (measured in meters squared, [math] m^2 [/math]).
• l is the length (measured in meters, m).
• Certain intrinsic characteristics of a substance determine its resistivity.
• It specifically has to do with how many free, conducting electrons can pass through the structure and how easily these electrons can move across it.
• This mobility is influenced by the temperature of the material, the distribution of impurities, and the configuration of the atoms in the conductor.
• Temperature also affects resistivity.
• As temperature rises, the resistivity of metals increases, while many semiconductors—including silicon and germanium—have a decreasing resistance.
• When a superconducting substance, such as a metal, is heated below its critical temperature, its resistivity increases with decreasing temperature, but it becomes zero below.

9. CORE PRACTICAL 2: Determine the electrical resistivity of a material.

• Materials:
  – A sample of the material (e.g., a wire or a block)- A multimeter (for measuring voltage and current)
  – A power source (e.g., a battery or a power supply)
  – Wires and connectors
  – A ruler or calipers (for measuring length and cross-sectional area)
• Procedure: 
  1. Measure the length (L) and cross-sectional area (A) of the material sample.
  2. Connect the material sample to the power source and multimeter in series.
  3. Apply a known voltage (V) across the material sample.
  4. Measure the current (I) flowing through the material sample.
  5. Repeat steps 3-4 for several different voltages.
  6. Plot a graph of V vs. I.
  7. Calculate the resistance (R) of the material sample using Ohm’s Law: [math] R = \frac{V}{I} [/math].
  8. Calculate the resistivity (ρ) of the material using the formula: [math] ρ = R * \frac{A}{L} [/math]
• Formula:
• [math] ρ = \fra{V}{I}  * \rac{A}{L} [/math]
• Where:
  – ρ is the electrical resistivity of the material
  – V is the voltage across the material
  – I is the current through the material
  – A is the cross-sectional area of the material
  – L is the length of the material
• Tips and Variations:
  – Use a constant current source for more accurate results.
  – Measure the temperature of the material, as resistivity can vary with temperature.
  – Use a four-probe method to eliminate contact resistance.
  – Compare your results with known values for the material.