Potential Difference and Energy Transfer in Series Circuits

Fundamentals of Electric Current and Energy Transfer

• Electric current is defined as a flow of electrons around an electrical circuit.
• Electron pathway in a closed circuit:
  • Electrons exit the power source (the cell).
  • They travel through the conducting wires.
  • They pass through the circuit components, such as a filament lamp.
  • They complete the continuous loop by returning to the cell.
• Mechanism of illumination:
  • A lamp illuminates because the electrons deliver and transfer energy to it as they pass through.
• Energy store transformations in a simple circuit:
  • The cell maintains a store of chemical energy.
  • This chemical energy is transformed into electrical energy, which is carried by the electrons flowing out of the cell.
  • As electrons pass through a component (e.g., a lamp), the electrical energy is transferred into other energy forms.
  • In a lamp, electrical energy is converted into light energy and thermal energy.

Definition and Principles of Potential Difference (Voltage)

• Potential difference is also referred to as voltage; both terms are interchangeable in scientific examinations.
• Conceptual purpose:
  • Potential difference describes and quantifies how energy is transferred within an electrical circuit.
• Quantitative definition:
  • A potential difference of $1\,\text{V}$ indicates that precisely $1\,\text{J}$ of energy is transferred for each $1\,\text{C}$ of charge moving through the circuit.
• General relation:   $\text{Potential Difference (V)} = \frac{\text{Energy (J)}}{\text{Charge (C)}}$

Measurement of Potential Difference Using a Voltmeter

• Instrument:
  • Potential difference is measured using a voltmeter.
  • The standard circuit symbol for a voltmeter is a circle enclosing the capital letter "V".
• Connection procedure:
  • A voltmeter is placed across the component or power source whose potential difference is being measured (connected in parallel).
• Single-component circuit measurements:
  • Measuring across a single cell: Placing a voltmeter across the cell yields a reading such as $9\,\text{V}$.
  • Physical meaning of a $9\,\text{V}$ cell reading: The electrons transfer $9\,\text{J}$ of energy per coulomb of charge flowing through the circuit.
  • Measuring across a single lamp: Placing a voltmeter across the lamp in this circuit also gives a reading of $9\,\text{V}$.
  • Energy implication: All of the electrical energy transported by the electric current is transferred into light energy and thermal energy within that single lamp.

Potential Difference in Series Circuits with Identical Components

• Circuit arrangement:
  • Two identical lamps connected in series across a single $9\,\text{V}$ cell.
• Total circuit energy transfer:
  • The potential difference across the power source remains $9\,\text{V}$.
  • The total energy transferred across the entire circuit is identical to that of a single-lamp circuit powered by the same cell.
• Observation of component brightness:
  • Both lamps in this series circuit are visibly dimmer than the single lamp in the single-lamp circuit.
• Distribution of potential difference:
  • Connecting a voltmeter across each lamp reveals a potential difference of $4.5\,\text{V}$ across each identical lamp.
  • Sum of potential differences across identical components:     $V_{\text{total}} = 4.5\,\text{V} + 4.5\,\text{V} = 9\,\text{V}$
• Explanation for reduced brightness:
  • In a single-lamp circuit, all energy carried by the current ($9\,\text{J}$ per coulomb) is transferred to the single lamp, making it shine brightly.
  • When two identical lamps are connected in series, the total energy carried by the current is shared equally between them.
  • Each lamp receives only $4.5\,\text{J}$ of energy per coulomb of charge, causing each lamp to glow dimmer.

Potential Difference in Series Circuits with Non-Identical Components

• Variance among components:
  • Components connected in series do not need to be identical and can have different potential differences.
• Voltage distribution across non-identical lamps:
  • In a series circuit powered by a $9\,\text{V}$ cell with two different lamps:
  • One lamp has a measured potential difference of $6\,\text{V}$.
  • The other lamp has a measured potential difference of $3\,\text{V}$.
  • Fundamental series rule:     $V_{\text{total}} = 6\,\text{V} + 3\,\text{V} = 9\,\text{V}$     The sum of the individual potential differences across all components in a series circuit is equal to the total potential difference provided across the power source (cell).
• Relative brightness of non-identical components:
  • The lamp with a potential difference of $6\,\text{V}$ is brighter than the lamp with a potential difference of $3\,\text{V}$.
  • Reason: The $6\,\text{V}$ lamp transfers $6\,\text{J}$ of energy per coulomb of charge, whereas the $3\,\text{V}$ lamp transfers only $3\,\text{J}$ of energy per coulomb of charge. The component that transfers more energy per unit charge operates at a higher brightness.
• Summary of energy conservation in series circuits:
  • A $9\,\text{V}$ cell provides $9\,\text{J}$ of energy for every coulomb of charge.
  • This energy is shared between all connected series components.
  • The total energy transferred across the individual components must balance the total energy supplied by the cell.