Topic3: Electric Circuits (Part 2)

Topic 3: Content
41)    Be able to use I = nqvA to explain the large range of resistivities of different materials
42)    Understand how the potential along a uniform current-carrying wire varies with the distance along it
43)    Understand the principles of a potential divider circuit and understand how to calculate potential differences and resistances in such a circuit
44)    Be able to analyse potential divider circuits where one resistance is variable including thermistors and light dependent resistors (LDRs)
45)    Know the definition of electromotive force (e.m.f.) and understand what is meant by internal resistance and know how to distinguish between e.m.f. and terminal potential difference
46)    CORE PRACTICAL 3: Determine the e.m.f. and internal resistance of an electrical cell.
47)    Understand how changes of resistance with temperature may be modelled in terms of lattice vibrations and number of conduction electrons and understand how to apply this model to metallic conductors and negative temperature coefficient thermistors
48)    Understand how changes of resistance with illumination may be modelled in terms of the number of conduction electrons and understand how to apply this model to LDRs.

1. Range of resistivity

  • A substance needs appropriate charge carriers, like as ions in electrolytes and gases or loosely bound electrons in metals, in order for a current to flow through it.
  • The experiment shown in Figure 1 illustrates the concept of a charge carrier.
  • The metallized sphere transfers charge as it swings back and forth between the metal plates when the high voltage is turned on.
  • Positive charge flows from left to right and vice versa, as seen in Figure 1b.
  • This is comparable to the circuit’s current flowing clockwise, which the nano ammeter (nA) records.
  • The sphere in this experiment might be thought of as the “charge carrier.”
  • Figure 1 Metallized polystyrene sphere acting as charge carrier.
  • It is common knowledge that a positive charge would flow in the same direction as an electric current. Should the carriers of charge be negative
  • The random thermal motion of charges, also known as thermal noise or Johnson-Nyquist noise. This is the random motion of charged particles, such as electrons, due to thermal energy.
  • At any temperature above absolute zero (-273°C or -459°F), particles in a conductor or semiconductor exhibit random motion due to thermal energy. This motion causes fluctuations in the electric current, even in the absence of an applied voltage.
  • Some key aspects of random thermal motion in charges:
    1. Temperature dependence: The amplitude of thermal noise increases with temperature.
    2. Randomness: The motion is unpredictable and random, making it a source of noise.
    3. Gaussian distribution: The thermal noise voltage follows a Gaussian distribution.
    4. White noise: Thermal noise is a type of white noise, meaning it has equal power density across all frequencies.
    5. Voltage fluctuations: Thermal noise causes small voltage fluctuations across a conductor or semiconductor.
    6. Current fluctuations: Similarly, it causes small current fluctuations.
  • Draft velocity is related to the drift velocity, which is the average velocity of charged particles in a conductor or semiconductor under the influence of an electric field.
  • Drift velocity is typically denoted by the symbol “vd” and is measured in units of meters per second (m/s).
  • Some key aspects of drift velocity in charges:
    1. Electric field: Drift velocity is proportional to the strength of the electric field applied to the conductor or semiconductor.
    2. Mobility: Drift velocity is also dependent on the mobility of the charged particles, which is a measure of how easily they move through the material.
    3. Scattering: Drift velocity is affected by scattering mechanisms, such as collisions with other particles or defects in the material.
    4. Current density: Drift velocity is related to the current density, which is the flow of charge per unit area.
  • Drift velocity is an important concept in understanding the behavior of charged particles in materials, particularly in electronic devices such as:
    1. Transistors: Drift velocity plays a crucial role in the operation of transistors, which are used to amplify or switch electronic signals.
    2. Diodes: Drift velocity is also important in diodes, which are used to rectify or regulate electronic signals.
    3. Solar cells: Drift velocity is relevant in solar cells, where it affects the efficiency of charge collection and conversion into electrical energy.
  • For a conductor, the current I is given by:
  • [math] I = nAvq [/math]
  • Where,
    A = area of cross-section of conductor
    n = number of charge carriers per cubic meter
    q = charge on each charge carrier
    v = drift velocity of charge carriers. 

     


    Figure 2 Direction of current

    This equation may be deduced as follows, with reference to Figure 2:
    ● Consider a section of wire of length Δx

  • ● The volume of wire in this section will be
  • [math] ΔV = AΔx [/math]
  • If there are n charge carriers per unit volume in the wire, the number in volume ΔV will be
  • [math] ΔN = nΔV = nAΔx [/math]
  • If the charge on each charge carrier is q, the quantity of charge within the section will be
  • [math] ΔQ = nAqΔx [/math]
  • Suppose each charge carrier takes a time Δt to travel the distance Δx.
  • Dividing both sides of the equation by Δt gives us
  • [math] \frac{\Delta Q}{\Delta t} = nAq \frac{\Delta x}{\Delta t} [/math]
  • ● [math] \frac{\Delta Q}{\Delta t} = I \quad \text{and} \quad \frac{\Delta x}{\Delta t} = v [/math]
  • ● Hence
  •   [math] I = nAqv [/math]

2. Potential along a uniform current-carrying wire:

  • A voltmeter is connected between two locations in a circuit to detect the potential difference between them.
  • Our discussion involves attaching a voltmeter in parallel or across a component to determine the potential difference between its ends.
  • The voltmeter in the circuit seen in Figure 3 is measuring the p.d. across the light.
  • The voltmeter would need to be connected between A and B to determine the p.d. across the resistor and between A and C to measure the p.d. across the cell.
  • Figure 3 Voltmeter measuring the p.d. across a lamp
  • A voltmeter must take some current in order to operate.
  • In the circuit in Figure 3, the ammeter records the circuit current (I) but the current through the lamp is only I – i, where i is the current taken by the voltmeter.
  • In order to keep i as small as possible, voltmeters should have a very high resistance.
  • The resistance of an object is dependent on it length
  • [math] R = \frac{ρl}{A} [/math]
  • Using ohm’s law
  • [math] V =IR [/math]
  • As resistance increases, potential will also increase.

    The potential along a uniform current-carrying wire increases uniformly with the distance along it.

3. Potential divider circuit :

  • A fundamental concept in electronics, used to reduce a voltage level or divide a voltage ratio.
  • A potential divider is a resistor network that divides an input voltage ([math] V_{in} [/math]) into a smaller output voltage ([math] V_{out} [/math]). It consists of two resistors, [math] R_1 \text{ and, } R_2 [/math] connected in series.
  • Working:
    1. The input voltage ([math] V_{in} [/math]) is applied across the series combination of [math] R_1 and R_2 [/math].
    2. The voltage across [math]R_2( V_{out}) [/math] is proportional to the ratio of [math] R_2 [/math]to the total resistance [math](R_1 + R_2) [/math].
    3. By adjusting [math]R_1 and R_2 [/math], the output voltage [math](V_{out}) [/math] can be set to a specific value.
  • Types:
    1. Fixed Potential Divider: Uses fixed resistors to divide the voltage.
    2. Variable Potential Divider: Uses a potentiometer (variable resistor) to adjust the output voltage.
  • The potential divider formula assumes ideal resistors and neglects loading effects.
  • In practice, the output voltage may vary due to resistor tolerances and loading.

  • Figure 4 Circuit diagram of a potential divider
  • Generally, we are trying to vary the potential difference (across ) by varying .
  • Assuming that the cell has negligible internal resistance, then the total resistance of the circuit  is given by:
  • [math] R_T = R_1 + R_2 [/math]
  • by using the Ohm’s law
  • [math] I = \frac{V}{R} = \frac{\varepsilon}{R_T} = \frac{\varepsilon}{R_1 + R_2} [/math]
  • Then considering the resistor R1 we can write:
  • [math] V_1 = IR_1 [/math]
  • Substituting:
  • [math] V_1 = \frac{\varepsilon}{R_1 + R_2} \cdot R_1[/math]
  • [math] V_1 = \frac{\varepsilon R_1}{R_1 + R_2} [/math]
  • From this equation it can be seen that if ε and [math] R_1[/math] are fixed, then [math] V_1 [/math] only depends on [math] R_2 [/math]. In fact, as [math] R_2 [/math] increases, [math] V_1 [/math] decreases, and vice versa.

4. Potential dividers as sensors:

  • Potential dividers can be used as sensors to measure various physical parameters, such as:
    1. Temperature: By using a thermistor or a temperature-dependent resistor in the potential divider circuit, the output voltage can be made to vary with temperature.
    2. Light: A photoresistor or light-dependent resistor can be used to measure light intensity.
    3. Pressure: A pressure-dependent resistor or a piezoresistor can be used to measure pressure changes.
    4. Position: A potentiometer can be used as a position sensor, where the output voltage indicates the position of the wiper.
    5. Force: A force-sensing resistor or a piezoresistor can be used to measure force or weight.
    6. Humidity: A humidity-dependent resistor can be used to measure humidity levels.
    7. pH: A pH-dependent resistor can be used to measure the acidity or basicity of a solution.
  • The potential divider sensor circuit typically consists of:
    1. A voltage source [math](V_{in}) [/math]
    2. A sensor resistor [math](R_{sense}) [/math] that changes value with the physical parameter being measured
    3. A fixed resistor [math] (R_{fixed}) [/math]
    4. A voltage output [math](R_{out}) [/math] that varies with the sensor resistance
  • By measuring the output voltage [math](R_{out}) [/math], the physical parameter can be inferred. Potential divider sensors are simple, low-cost, and widely used in various applications.

  • Figure 5 A thermistor potential divider circuit and the resistance–temperature graph for a ntc thermistor
  • [math]R_2 [/math] is replaced by a thermistor coupled to an electronic thermometer.
  • Negative temperature coefficient, or NTC, thermistors make up the majority of thermistors.
  • This indicates that, as the temperature rises, their resistance falls, as seen in Figure 12, which also includes a circuit schematic illustrating the connections between the parts:
  • As the temperature rises, the resistance of the thermistor,[math] R_2 [/math] , decreases and [math]V_1 [/math] increases, which has an impact on the potential difference across the fixed resistor [math]R_1 [/math] in figure 11.
  • As a result, rising temperatures lead to rising pd. Conversely, if the voltmeter is connected across the thermistor, a rise in temperature will result in a fall in pd.
  • The voltmeter is often connected across the fixed resistor since most applications call for the pd to rise with temperature.
  • Using a light-dependent resistor yields a comparable result (LDR).
  • These parts serve as excellent light sensors since they alter in resistance in response to changes in light intensity.
  • Since semi-conducting materials are used to make LDRs, light may flow through them and release electrons from their structural bonds, lowering the LDR’s resistance.
  • LDRs can have resistances as high as megaohms while operating in the dark, but they can also have resistances as low as a few hundred ohms when operating in the light.
  • When an LDR is used in place of [math] R_2 [/math] in a potential divider circuit, the output voltage  across the fixed resistor will grow in tandem with the intensity of the light.

  • Figure 6 An LDR: its electrical circuit symbol (a) resistance– light intensity graph (b) and its use in a potential divider circuit (c).

5. Internal resistance and electromotive force:

  • Electromotive Force (EMF):
    – EMF is the voltage generated by a cell or battery when no current is flowing through it.
    – Measured in volts (V), it’s the “open-circuit voltage” of a cell or battery.
    – EMF is the maximum voltage a cell or battery can provide.
  • Internal Resistance:
    – Internal resistance is the opposition to current flow within a cell or battery.
    – Measured in ohms (Ω), it depends on the cell’s or battery’s chemistry, age, and other factors.
    – Internal resistance causes a voltage drop when current flows, reducing the available voltage.
  • Relationship between EMF and Internal Resistance:
    – When current flows, the internal resistance reduces the voltage available from the EMF.
    – The higher the internal resistance, the greater the voltage drop.
    – The lower the internal resistance, the closer the available voltage is to the EMF.
  • Internal resistance, or r, is a constant feature of real power sources, such batteries and lab power packs.
  • The power supply’s internal resistance produces a potential difference as current passes through it, which causes electrical energy to be converted to heat energy.
  • This is one of the causes of the warming up of portable electronics like tablets after extended usage.
  • Since internal resistance is located “inside” a power source, it cannot be tested directly.
  • It can only be measured by applying its electrical characteristic.
  • Figure 7 illustrates a circuit that may be used to do this.

  • Figure 7 A circuit used to measure the internal resistance and electromotive force of a real power supply and a real power supply
  • According to Kirchoff’s Second Circuit Law, the electromotive force, ε, must equal the total of the potential differences in the circuit.
  • In this circuit, there are two potential differences: the pd across the internal resistor and the one across the external variable resistor (V).
  • The pd across this resistor is equal to Ir, however this cannot be determined precisely.
    This implies that:
  • [math]\varepsilon = V + Ir[/math]
  • The current, I, can be measured directly using an ammeter; ε and r are both constants, so the equation can be rewritten as:
  • [math]V = \varepsilon – Ir[/math]
  • Or
  • [math]V = -rI + \varepsilon [/math]
  • It is the equation for a straight line with a negative gradient that looks like this: y = mx + c.
  • The electromotive force, ε, is the graph’s y-intercept, and the gradient is negative and equal to –r, the internal resistance, if an electrical characteristic is created using values of V and I from different values of R (the external load resistance)

6. CORE PRACTICAL 3: Determine the e.m.f. and internal resistance of an electrical cell.

  • Apparatus:
    – Electrical cell (e.g., battery)
    – Variable resistor (e.g., rheostat)
    – Voltmeter
    – Ammeter
    – Connecting wires
  • Procedure:
    1. Setup: Connect the cell, variable resistor, voltmeter, and ammeter in a circuit, as shown:


      Figure 8 Determine the e.m.f. and internal resistance of an electrical cell.

  • Cell → Variable Resistor → Voltmeter → Ammeter → Cell
    1.  Measurements:
      – Vary the resistance using the variable resistor.
      – Take multiple readings of the voltage (V) across the cell and the current (I) flowing through the circuit.
      – Record your data in a table.
    2. Data analysis:
      – Plot a graph of voltage (V) vs. current (I).
      – The graph should be a straight line.
      – The slope of the line represents the internal resistance (r) of the cell.
      – The y-intercept (where the line crosses the voltage axis) represents the e.m.f. (ε) of the cell.
  • Calculations:
    – Internal resistance (r): Calculate the slope of the graph using the formula: [math] r = ΔV / ΔI [/math]
    – e.m.f. (ε): Read the y-intercept value from the graph.
  • Tips and Variations:
    – Use a range of resistances to ensure accurate results.
    – Repeat the experiment with different cells or types of cells (e.g., alkaline, nickel-cadmium).
    – Measure the temperature of the cell during the experiment to investigate its effect on e.m.f. and internal resistance.
  • By following this practical, you’ll determine the e.m.f. and internal resistance of an electrical cell, gaining hands-on experience with circuit measurements and data analysis.

7. Modeling changes in resistance:

  • The resistance of a material can be modeled using the equation:
  • [math] R = \frac{ρ * L}{A} [/math]
  • where:
    – R is the resistance
    – ρ (rho) is the resistivity of the material
    – L is the length of the material
    – A is the cross-sectional area of the material
  • When light is incident on a material, it can excite electrons, increasing the number of conduction electrons (n).
  • This reduces the resistivity (ρ) of the material, leading to a decrease in resistance (R).
  • Lattice Vibrations (Phonons):
  • As temperature increases, the lattice vibrations (phonons) in the material increase, causing the electrons to scatter more frequently. This increases the resistivity (ρ) and therefore the resistance (R).
  • Conduction Electrons and Resistance:
  • The number of conduction electrons (n) is related to the resistivity (ρ) by:
  • [math] ρ ∝ 1/n [/math]
  • As the number of conduction electrons (n) increases, the resistivity (ρ) decreases, leading to a decrease in resistance (R).
  • Applying the Model to LDRs:
  • LDRs (Light Dependent Resistors) are made from materials whose resistivity changes significantly with light intensity.
  • The model above can be applied to LDRs as follows:
    – In the dark, the number of conduction electrons (n) is low, resulting in high resistivity (ρ) and high resistance (R).
    – When light is incident on the LDR, the number of conduction electrons (n) increases, reducing resistivity (ρ) and resistance (R).
    – The resistance of the LDR decreases with increasing light intensity.
  • LDR Characteristics:
  • LDRs have the following characteristics:
    – High resistance in the dark (typically kΩ to MΩ).
    – Low resistance in bright light (typically Ω to kΩ).
    – Resistance decreases exponentially with increasing light intensity.
  • By understanding how changes in resistance with illumination can be modeled in terms of the number of conduction electrons, you can apply this model to LDRs and predict their behavior in various lighting conditions. This knowledge is essential for designing and working with LDR-based circuits.
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