DC Circuit

The Battery and EMF (Electromotive Force)

What is Electromotive Force (EMF)

• Definition: Electromotive force, or emf, is the energy required to move a unit electric charge by an energy source such as a battery, cell, or generator.
  • Characteristics: It is defined as the potential difference across the terminals where there is no current passing through it, exemplifying an open circuit situation wherein one terminal is positive and the other is negative.

Nature of EMF

• Misconception: In reality, the electromotive force is not a force but a measure of energy.
• Energy Conversion: The source converts one form of energy into electrical energy:
  • Battery: Converts chemical energy.
  • Generator: Converts mechanical energy.
  • Solar Cells: Create voltage directly from light.
  • Thermoelectric Devices: Generate voltage from temperature differences.

Examples of Voltage Sources

• The output voltage from these sources is contingent upon their construction and load and equates to emf only when unloaded.
  • Notable Examples:
  • Brazos Wind Farm (Fluvanna, Texas)
  • Krasnoyarsk Dam (Russia)
  • Solar farms documented by the U.S. Department of Energy
  • Nickel-metal hydride batteries

Lead-Acid Battery

• The lead-acid battery is one of the most commonly utilized types in vehicles.
  • Components: A single cell consists of:
  • Anode (positive terminal): Connected to a lead oxide plate.
  • Cathode (negative terminal): Connected to a lead plate.
  • Both plates are submerged in sulfuric acid, which serves as the electrolyte.
  • Chemical Processes:
  • Electrons are chemically deposited at the lead plate (Pb), rendering it negatively charged.
  • Electrons are chemically removed from the lead oxide plate (PbO2), making it positively charged.

Electromotive Force Equation

• The relationships involved in an electric circuit are expressed using the following variables:

  • I = Current flowing through the circuit
  • R = Resistance of the resistor
  • r = Internal resistance of the battery

• Key Equations:

  • Voltage across resistor (V): $V = IR$
  • Total voltage (E): $E = I(R + r)$

Voltage Sources and Circuits

Series Connections

• Voltage sources can be connected in series to produce a higher total electromotive force.
• When sources are connected in series, the internal resistances add along with their emf values, giving:
  • $E<em>{ ext{total}} = E</em>1 + E_2$
• Applications: Common in flashlights, toys, and various electrical appliances.

Disadvantages of Series Connections

• Internal Resistance: The internal resistances also add, which can limit current output.

Opposing Voltage Sources

• If cells are connected in such a way that they oppose one another (e.g., one battery is reversed), then:
  • The total emf becomes the algebraic sum of individual emfs, which may lead to a lower total output.
  • Current flows in the direction of the higher emf and is limited by the sum of the internal resistances.
  • Example: A battery charger must have a larger emf than the charged battery to operate effectively and reverse current flow.

Parallel Connections

• When voltage sources are connected in parallel:
  • The total emf remains equal to the individual source emfs.
  • Internal resistances are reduced since the resistances are arranged in parallel, thus allowing for higher current output.

EMF vs. Voltage

Definitions

• EMF (Electromotive Force):

  • Symbol: $ext{ε}$
  • Formula: $ext{ε} = V + IR$
  • Meaning: Amount of energy supplied by the source to a unit charge.
  • Measurement: Measured between the endpoints of the source when no current is flowing (open circuit).
  • Device: Measured using an EMF meter.

• Voltage (V):

  • Formula: $V = IR$
  • Meaning: Energy used by a unit charge to move from one point to another in a circuit.
  • Measurement: Measured across a resistor when current is flowing (closed circuit).
  • Device: Measured using a voltmeter.

Comparison

• Magnitude: EMF is greater than the voltage across a resistor under load conditions.

Power Relations

• Power Equations:
  • Power dissipated by the battery: $P_{ ext{battery}} = I^2r$
  • Power dissipated by the resistor: $P_{ ext{resistor}} = I^2R$
  • Power supplied by the EMF: $P = I ext{ε} = I^2R + I^2r$

Series Circuits and Current Behavior

• In series circuits:
  • Current remains constant across all components.
  • Effective voltage is directly proportional to the total resistance.
  • The applied voltage is equal to the sum of the voltage drops across each element.

Parallel Circuits and Current Behavior

• In parallel circuits:
  • Voltage drops are equal across all components.
  • The current flowing through each component is inversely proportional to its resistance.
  • Total circuit current is the sum of the currents through each parallel path.
  • The reciprocal of total resistance is equal to the sum of the reciprocals of individual resistances:
  • rac{1}{R{ ext{total}}} = rac{1}{R1} + rac{1}{R2} + rac{1}{R3}

Kirchhoff's Laws

• Kirchhoff's Current Law (KCL): The sum of currents entering a junction is equal to the sum of currents leaving the junction. Mathematically expressed as:

  • $I<em>1 + I</em>2 + I<em>3 - (I</em>4 + I_5) = 0$

• Kirchhoff's Voltage Law (KVL): The algebraic sum of all voltages in a closed circuit is zero:

  • $V<em>{AB} + V</em>{BC} + V<em>{CD} + V</em>{DA} = 0$

Application of Kirchhoff's Laws

• Kirchhoff's laws are used to analyze complicated circuits that cannot be reduced to simpler series or parallel forms.

Measurement Instruments

Ammeters

• Function: Measure the current in a wire.
• Connection: Must be placed in series with the component being measured.
• Ideal Characteristics: Should have negligible resistance to avoid altering the current.

Voltmeters

• Function: Measure the voltage across components in a wire.
• Connection: Must be placed in parallel with the component being measured.
• Ideal Characteristics: Should have very high resistance to prevent current from flowing through them while taking a measurement.