D.C. Circuit 

• a) The idea that the current from a source is equal to the sum of the currents in the separate branches of a parallel circuit, and that this is a consequence of conservation of charge

• This principle is derived from Kirchhoff’s Current Law (KCL) and is rooted in the law of conservation of charge, which states that charge cannot be created or destroyed in an isolated system.
• ⇒ Explanation:
• When current flows in an electrical circuit, it is the flow of charges (usually electrons). At any junction (or node) in a parallel circuit, the total current entering the junction must equal the total current leaving the junction.
• This is because the number of charges flowing into the junction per unit time must equal the number of charges flowing out to maintain charge conservation.
• ⇒ Mathematical Representation:
• If the current from the source is [math]I_{source}[/math]​, and the currents in the branches of the parallel circuit are ​[math]I_1, I_2, I_3, …, …. [/math], then:
• [math]I_{source} = I_1, I_2, I_3, …, ….[/math]
• 
• Figure 1 Kirchhoff’s current law
• ⇒ Practical Implication:
• For example, if a 3 A current flows from a source into a parallel circuit with two branches, where branch 1 has a current of 2 A and branch 2 has a current of 1 A, the sum of the branch currents equals the source current (2 A + 1 A = 3 A). This satisfies conservation of charge.

• b) The sum of the potential differences across components in a series circuit is equal to the potential difference across the supply, and that this is a consequence of conservation of energy

• This principle is based on Kirchhoff’s Voltage Law (KVL) and the law of conservation of energy, which states that energy cannot be created or destroyed, only transformed or transferred.
• 
• Figure 2 Kirchhoff’s Voltage Law (KVL)
• ⇒ Explanation:
• In a series circuit, the electrical energy supplied by the source (e.g., a battery or power supply) is distributed across all components (resistors, bulbs, etc.).
• The total energy per unit charge provided by the source (potential difference across the source) must equal the sum of the energy per unit charge used by each component (potential differences across the components).
• ⇒ Mathematical Representation:
• If the potential difference across the supply is [math]V_{total}[/math]​, and the potential differences across the series components are [math]V_1 + V_2 + V_3, …[/math] then:
• [math]V_{total} = V_1 + V_2 + V_3, …[/math]
• ⇒ ​Practical Implication:
• For instance, if a 12 V battery powers a series circuit with two resistors where [math]V_1 = 7V[/math] and [math]V_1 = 5V[/math], then the total voltage supplied is [math]12V (V_1 + V_2 = 12V)[/math].
• This is consistent with conservation of energy because the work done on the charges by the battery is fully accounted for by the energy dissipated in the resistors.

• c) Potential differences across components in parallel are equal

• This principle is a direct consequence of how potential difference (voltage) behaves in a parallel circuit.
• ⇒ Explanation:
• In a parallel circuit, all components are connected across the same two points (or nodes).
• Since potential difference is defined as the energy difference per unit charge between two points, every branch in the parallel circuit experiences the same potential difference because they share the same two points.
• ⇒ Mathematical Representation:
• If the potential difference across the supply is [math]V_{source}[/math]​, then the potential difference across each parallel branch ([math]V_1, V_2,  V_3, …[/math]is the same:
• [math]V_{source} = V_1 = V_2 = V_3 = …,[/math]
• 
• Figure 3 Potential difference across the source
• ⇒ Practical Implication:
• If a 9 V battery is connected to a parallel circuit with three branches, then the voltage across each branch is 9 V, regardless of the components in each branch. This ensures that each branch receives the same “push” of electrical energy per unit charge.
• ⇒ Relationship to Conservation Laws
• 1. Conservation of Charge:
• Ensures that the total current entering and leaving a node (as in point a) is consistent because the charges flowing into the system must balance those flowing out.
• 2. Conservation of Energy:
• Ensures that the total energy supplied by the source (as in points b and c) is fully accounted for in the circuit, either by the components in series or by maintaining equal potential differences across branches in parallel.
• These principles form the foundation of circuit analysis and are essential for understanding how electrical circuits function.

• d) The application of equations for the combined resistance of resistors in series and parallel

• The combined (or equivalent) resistance of resistors depends on whether the resistors are arranged in series or parallel.
• ⇒ Resistors in Series:
• When resistors are connected in series, the total resistance is simply the sum of the individual resistances.
• ⇒ Explanation:
• In a series circuit, the same current flows through all resistors, but the potential difference across each resistor is different.
• The total potential difference across the combination is equal to the sum of the potential differences across each resistor. The resistances add up because the total opposition to the current is cumulative.
• 
• Figure 4 Resistance in Series
• ⇒ Equation:
• For resistors [math]R_1, R_2 [/math], and ​[math]R_3[/math] in series:
• [math]R_{total} = R_1 + R_2 + R_3[/math]
• ⇒ Practical Implication:
• If three resistors with resistances of 2 Ω, 3 Ω, and 5 Ω are connected in series, the total resistance is:
• [math]R_{\text{total}} = 2 + 3 + 5 \\
  R_{\text{total}} = 10\ \Omega [/math]
• ⇒ Resistors in Parallel
• When resistors are connected in parallel, the total resistance is less than the resistance of the smallest resistor.
• ⇒ Explanation:
• In a parallel circuit, the voltage across each resistor is the same, but the current splits between the branches.
• The total resistance decreases because the circuit provides multiple paths for the current to flow, reducing the overall opposition.
• 
• Figure 5 Resistors in Parallel
• ⇒ Equation:
• For resistors [math]R_1, R_2[/math]​, and [math]R_3[/math]​ in parallel:
• [math]\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}[/math]
• or for two resistors:
• [math]R_{\text{total}} = \frac{R_1 R_2}{R_1 + R_2}[/math]
• ⇒ Practical Implication:
• If two resistors with resistances of 6 Ω and 3 Ω are connected in parallel:
• [math]\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \\
  \frac{1}{R_{\text{total}}} = \frac{1}{6} + \frac{1}{3} \\
  \frac{1}{R_{\text{total}}} = \frac{3}{6} \\
  R_{\text{total}} = 2\ \Omega [/math]

• e) The use of a potential divider in circuits (including circuits with LDRs and thermistors)

• ⇒ Potential Divider:
• A potential divider is a circuit used to divide the voltage from a power supply into smaller, desired values. It typically consists of two resistors connected in series. The output voltage is taken across one of the resistors.
• 
• Figure 6 Potential Divider
• ⇒ Basic Principle
• The voltage across a resistor in a series circuit is proportional to its resistance. Using the potential divider equation:
• [math]V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2}[/math]
• Where:
• – [math]V_{\text{in}}[/math] is the supply voltage,
• – [math]V_{\text{out}}[/math] is the output voltage,
• – [math]R_1, R_2[/math] are the resistances.
• ⇒ Potential Dividers with LDRs and Thermistors
• LDR (Light-Dependent Resistor):
• An LDR’s resistance decreases as light intensity increases. In a potential divider circuit:
• – If the LDR is in the top part of the divider ([math]R_1[/math]​), [math] V_{out}[/math] decreases as light increases.
• – If the LDR is in the bottom part of the divider ([math]R_2[/math]), [math]V_{out}[/math]​ increases as light increases.
• 
• Figure 7 Potential divider with LDR
• ⇒ Thermistor:
• A thermistor’s resistance decreases as temperature increases (for an NTC thermistor). In a potential divider circuit:
• – If the thermistor is in the top part of the divider ([math]R_1[/math]​), [math]V_{out}[/math] ​ decreases as temperature increases.
• – If the thermistor is in the bottom part of the divider ([math]R_1[/math]), [math]V_{out}[/math] ​ increases as temperature increases.
• 
• Figure 8 Potential divider with thermistor
• ⇒ Applications
• LDR Circuits:
• – Used in automatic lighting systems where V_out ​ can control a switch to turn lights on or off based on light levels.
• Thermistor Circuits:
• – Used in thermostats or temperature control systems.

• f) The emf of a source:

• ⇒ Definition
• The electromotive force (emf) of a source is the total energy per unit charge provided by the source. It represents the maximum potential difference the source can supply when no current is flowing (open circuit condition).
• ⇒ Explanation
• The emf is typically measured in volts (V) and is often labeled .
• It accounts for the energy supplied to move charges around the circuit, including overcoming the internal resistance of the source.
• ⇒ Equation for emf
• If a source has internal resistance r, and a current I flows through the circuit with an external resistance R, the terminal voltage V (voltage measured across the source) is:
• [math]V = ε – Ir[/math]
• Where:
• – [math]ε[/math]is the emf,
• – Ir is the voltage drop due to the internal resistance.
• 1. Open Circuit:
• When no current flows, I=0, so the terminal voltage equals the emf (V= [math]ε[/math]).
• 2. Closed Circuit:
• When current flows, the terminal voltage is less than the emf due to the internal resistance (V<ε ).
• ⇒ Practical Implications
• A battery labeled “12 V” typically has an emf of 12 V. However, if current flows and the battery has internal resistance, the terminal voltage may drop below 12 V.
• Emf is a fundamental quantity in energy sources, including batteries, solar cells, and generators, as it determines the source’s ability to do work on charges.
• These principles form the foundation of understanding resistive circuits, voltage control, and energy sources in electronics.

• g) The unit of emf is the volt (V), which is the same as that of potential difference

• ⇒ Electromotive Force (emf) and Potential Difference
• Electromotive force (emf):
• – Emf refers to the total energy supplied by a source to move 1 coulomb of charge around a circuit. It represents the maximum potential difference a source can provide when no current flows.
• Potential Difference:
• – Potential difference (voltage) between two points in a circuit is the work done to move 1 coulomb of charge between those points.
• ⇒ Similarities and Differences
• 1. Unit:
• Both emf and potential difference are measured in volts (V), where:
• [math]V = 1 \text{joule per coulomb} \ (J/C)[/math]
• 2. Context:
• – Emf is the energy provided by the source. It includes energy losses due to internal resistance.
• – Potential difference is the energy used by the charges as they pass through components (e.g., resistors or bulbs).

• h) The idea that sources have internal resistance and to use the equation

• ⇒ Internal Resistance
• Every real power source (e.g., a battery or generator) has some internal resistance r due to the materials it is made of.
• Internal resistance causes energy loss within the source itself, as some energy is converted into heat.
• ⇒ Equation for a Source with Internal Resistance
• The relationship between emf ([math]ε[/math]), terminal voltage (V), current (I), and internal resistance (r) is given by:
• [math]V = ε – Ir[/math]
• Where:
• – V: Terminal voltage (measured across the source’s terminals)
• – [math]ε[/math]: Emf of the source,
• – I: Current flowing through the circuit,
• – r: Internal resistance of the source.
• ⇒ Explanation
• 1. When no current flows (I=0):
• The terminal voltage equals the emf:
• [math]V = ε[/math]
• 2. When current flows (I>0):
• The terminal voltage drops below the emf due to the energy loss in the internal resistance:
• [math]V = ε – Ir[/math]
• ⇒ Example:
• If a battery has an emf of 12 V, an internal resistance of 1 Ω, and it supplies a current of 2 A:
• [math]V = \mathcal{E} – I r \\
  V = 12 – (2 \times 1) \\
  V = 10\ V[/math]
• The terminal voltage is 10 V, meaning 2 V is lost within the battery.

• i) How to calculate current and potential difference in a circuit containing one cell or cells in series

• 1. Single Cell in a Circuit
• When a single cell with emf ([math]ε[/math]) and internal resistance (r) powers a circuit with an external resistance (R):
• Total Resistance ( [math]R_{total}[/math]​):
• [math]R_{total} = R + r[/math]
• Current in the Circuit (I): Using Ohm’s Law:
• [math]I = \frac{\mathcal{E}}{R_{\text{total}}} \\ I = \frac{ε}{R + r }[/math]
• Terminal Voltage (V):
• [math]V = IR \\ V = ε – Ir[/math]
• ⇒ Example:
• A cell with [math]ε[/math] = 9V, r = 1 Ω, connected to an external resistor R = 4 Ω:
• 1. Total resistance:
• [math]R_{\text{total}} = R + r \\
  R_{\text{total}} = 4 + 1 \\
  R_{\text{total}} = 5\ \Omega [/math]
• 2. Current:
• [math]I = \frac{\mathcal{E}}{R_{\text{total}}} \\
  I = \frac{9}{5} \\
  I = 1.8\ \text{A} [/math]
• 3. Terminal voltage:
• [math]V = IR \\ V = (1.8)(4) \\ V = 7.2 V[/math]
• 2. Multiple Cells in Series
• When cells are connected in series, their emfs and internal resistances add up.
• Total Emf ([math]ε_{total} [/math] ​):
• [math]ε_{total} = ε_1 + ε_2 + ε_3 + ⋯ + ε_n[/math]
• Total Internal Resistance ( ​[math]r_{total}[/math]):
• [math]r_{total} = r_1 + r_2 + r_3 + ⋯ + r_n[/math]
• 
• Figure 9 Internal resistance with emf
• Current (I):
• [math]I = \frac{\mathcal{E}_{\text{total}}}{R + r_{\text{total}}}[/math]
• ⇒ Example:
• Three cells, each with E = 6 V and r = 0.5 Ω, are connected in series to an external resistor R = 10 Ω:
• 1. Total emf:
• [math]\mathcal{E}_{\text{total}} = 6 + 6 + 6 \\
  \mathcal{E}_{\text{total}} = 18\ \text{V}[/math]
• 2. Total internal resistance:
• [math]r_{\text{total}} = 0.5 + 0.5 + 0.5 = 1.5\ \Omega[/math]
• 3. Total resistance:
• [math]R_{\text{total}} = R + r_{\text{total}} \\
  R_{\text{total}} = 10 + 1.5 \\
  R_{\text{total}} = 11.5\ \Omega[/math]
• 4. Current:
• [math]I = \frac{\mathcal{E}_{\text{total}}}{R_{\text{total}}} \\
  I = \frac{18}{11.5} \\
  I \approx 1.57\ \text{A}[/math]
• 5. Terminal voltage:
• [math]V = I R \\
  V = (1.57)(10) \\
  V = 15.7\ \text{V}[/math]
• Single Cell: The current depends on both the external resistance and the internal resistance.
• Multiple Cells in Series: The total emf and internal resistance are the sums of individual values.
• Terminal Voltage: Always less than the emf if current flows, due to the internal resistance.

Specified Practical Work

• Determination of the Internal Resistance of a Cell

• To determine the internal resistance of a cell, we can use a straightforward experimental setup that involves measuring the terminal voltage and current for various external resistances. Here’s a detailed step-by-step explanation of the experiment:
• ⇒ Principle
• The internal resistance (r) of a cell is the resistance within the cell that causes a drop in its terminal voltage when current flows. The relationship is given by:
• [math]V = ε – Ir[/math]
• Where:
• – V: Terminal voltage (volts),
• – [math]ε[/math]: Emf of the cell (volts),
• – I: Current (amperes),
• – r: Internal resistance (ohms).
• By varying the current (I) using different external resistances (R) and measuring V, we can calculate r.
• ⇒ Apparatus
  1. A cell or battery (whose internal resistance is to be determined).
  2. Variable resistor or a set of known resistors.
  3. Digital or analog voltmeter (to measure terminal voltage).
  4. Ammeter (to measure current).
  5. Connecting wires and crocodile clips.
  6. Switch (to control the circuit).
• 
• Figure 10 Internal Resistance of a cell
• ⇒ Circuit Diagram
• The circuit consists of:
• – The cell connected in series with an external resistor (R)
• – An ammeter in series to measure current (I),
• – A voltmeter connected across the cell to measure the terminal voltage (V).
• ⇒  Procedure
• 1. Set Up the Circuit:
• Connect the components as shown in the circuit diagram. Ensure the connections are tight and the polarities of the ammeter and voltmeter are correct.
• 2. Measure the Emf ([math]ε[/math]) of the Cell:
• Before connecting the variable resistor, measure the open-circuit voltage of the cell (no current flows). This is the emf ([math]ε[/math] ).
• 3. Introduce an External Resistance:
• Set the variable resistor to a high resistance value (or connect a known external resistor). Close the switch to allow current to flow.
• 4. Record Readings:
• Measure and record the terminal voltage (V) using the voltmeter and the current (I) using the ammeter.
• 5. Vary the Resistance:
• Adjust the variable resistor to change the current flowing in the circuit. For each setting, record the terminal voltage (V) and the current (I).
• 6. Repeat the Measurements:
• Take readings for at least 5–7 different values of I and V. Ensure the current values are within safe limits for the cell to avoid overheating or damaging it.
• 7. Switch Off the Circuit:
• Turn off the switch between measurements to prevent the cell from discharging excessively.
• ⇒ Data Analysis
• 1. Equation Rearrangement:
• From the equation[math]V = ε – Ir,[/math] rearrange it to:
• [math]V = ε – Ir[/math]
• This is in the form of a straight-line equation:
• [math]y = mx + c,[/math]
• where:
• – V is the dependent variable (y)
• – I is the independent variable (x),
• – −r is the slope (m),
• – [math] ε[/math] is the y-intercept (c).
• 2. Graph Plotting:
• Plot a graph of V (terminal voltage) on the y-axis against I (current) on the x-axis.
• 3. Determine the Slope and Intercept:
• The slope of the line (−r) gives the negative internal resistance. From this, calculate r.
• The y-intercept of the graph gives the emf ([math] ε[/math] ).
• ⇒ Example Calculation
• ⇒ Experimental Data:

• ⇒ Graph Analysis:
1. Plot V I.
2. Draw the best-fit straight line.
3. Determine the slope (−r) from the graph:

[math]r = – \frac{\Delta V}{\Delta I}[/math]

For example, if the slope is −0.5 Ω, then the internal resistance is r=0.5 Ω.

4. Determine the y-intercept to find the emf ([math]ε[/math] ).
• ⇒ Precautions
1. Ensure all connections are secure to minimize contact resistance.
2. Use a sensitive ammeter and voltmeter for accurate measurements.
3. Avoid drawing too much current from the cell to prevent excessive heating or damage.
4. Record measurements quickly to avoid changes due to cell discharge.
• ⇒ Sources of Error
1. Instrument Errors: Limited accuracy of voltmeter or ammeter.
2. Contact Resistance: Resistance at connections may affect measurements.
3. Temperature Effects: Internal resistance may change as the cell heats up during the experiment.
• ⇒ Conclusion
• ⇒ From the experiment:
1. The slope of the V–I graph provides the internal resistance (r).
2. The y-intercept of the graph gives the emf ([math]ε[/math]) of the cell.