Electromagnetic Induction and Electromotive Force Study Guide

Course Information and Learning Objectives

• Course Title: PHYSICS 121: General Physics III (Electromagnetic Induction)

• Credit Units: 2 Units

• Lecturer: Dr. Okafor P.N.

• Department: Department of Physics, College of Science and Technology, Covenant University

• Academic Session: 2024/2025

Student Learning Objectives

By the end of this lecture, students should be able to:

• Explain the term "electromotive force" (EMF).

• Explain high-level concepts of "electromagnetic induction."

• State the conditions necessary to increase or decrease induced EMF.

• Explain the concept of self-induction.

• Explain the concept of mutual induction.

Electromotive Force (EMF) and Potential Difference

• Definition of EMF: Electromotive force is a measure of electric potential difference that drives electrical current. It is measured in volts ($V$). Despite the name "force," it does not refer to a physical mechanical force.

• Static Definition: EMF can be defined as the potential difference across the terminals of a cell when it is not delivering current (open circuit conditions).

• EMF vs. Terminal Voltage:     - Electromotive Force (emf): The potential difference of a source when no current is flowing ($I = 0$).     - Terminal Voltage ($V$): The voltage output of a device measured across its terminals when it is part of a closed circuit.

• Mechanism of Flow: Electric potential difference creates an electric field. This field exerts a force on charges, which causes the flow of electric current.

• Energy Perspective: EMF is the energy per unit electric charge imparted by an energy source (like a battery or generator). As the device does work on the electric charge being transferred within itself, energy is converted from one form (chemical or mechanical) to electrical. This work done per unit of electric charge results in one terminal becoming positively charged and the other negatively charged.

• Unit of EMF: The unit is the Volt ($V$). If one coulomb ($C$) of charge is driven through a potential difference of one volt, the work done is one joule ($J$).

• Work-Energy Formula: $\text{Electrical work in Joules} = \text{Charge} \times \text{Voltage}$

Circuit Equations and Internal Resistance

• The EMF Equation: The total EMF ($E$) of a cell must account for both the external circuit and the internal resistance of the cell itself. $E = I \times (R + r)$

• Variables:     - $E$ = Electromotive Force ($V$)     - $I$ = Current flowing through the circuit ($A$)     - $R$ = External resistance ($\Omega$)     - $r$ = Internal resistance of the cell ($\Omega$)

• Terminal Potential Difference ($V$): This is the voltage across the external resistor. $V = I \times R$

• Voltage Drop due to Internal Resistance: $v = I \times r$

• Combined Relation: $E = V + (I \times r)$

Worked Examples: EMF and Resistance

• Example 1:     - Problem: A circuit has an external resistor of $5\,\Omega$ and a cell with internal resistance of $0.5\,\Omega$. If the current flowing is $1\,A$, calculate the EMF of the cell.     - Solution: $E = I(R + r)$ $E = 1 \times (5 + 0.5)$ $E = 5.5\,\text{volts}$

• Example 2:     - Problem: A cell with unknown EMF $E$ and internal resistance $2\,\Omega$ is connected to a $5\,\Omega$ resistor. The terminal potential difference ($V$) is measured as $1\,V$. Calculate $E$.     - Solution Step 1 (Calculate Current): Since Terminal PD ($V$) is across the $5\,\Omega$ resistor: $I = \frac{V}{R} = \frac{1}{5} = 0.2\,A$     - Solution Step 2 (Calculate Internal PD): PD across internal resistance $r$ is: $V_{internal} = I \times r = 0.2 \times 2 = 0.4\,V$     - Solution Step 3 (Calculate EMF): $E = V + V_{internal} = 1 + 0.4 = 1.4\,V$     - Alternative Solution: $E = I(R + r) = 0.2 \times (5 + 2) = 0.2 \times 7 = 1.4\,V$

• Class Drill:     - Problem: A wire of $4\,\Omega$ is connected to a battery of EMF $6\,V$ and internal resistance $r$. The current is $1\,A$. Calculate $r$.     - Solution: $E = I(R + r) \rightarrow 6 = 1(4 + r)$ $6 = 4 + r \rightarrow r = 2\,\Omega$

• Example 3:     - Problem: The terminal PD of a battery is $12\,V$ when an external resistance of $20\,\Omega$ is connected, and $13.5\,V$ when an external resistance of $45\,\Omega$ is connected. Calculate the EMF ($E$) of the battery.     - Solution Step 1 (Case 1): Current $I_1 = \frac{12}{20} = 0.6\,A$. Equation: $E = 12 + 0.6r$     - Solution Step 2 (Case 2): Current $I_2 = \frac{13.5}{45} = 0.3\,A$. Equation: $E = 13.5 + 0.3r$     - Solution Step 3 (Solve for r): $12 + 0.6r = 13.5 + 0.3r$ $0.3r = 1.5 \rightarrow r = 5\,\Omega$     - Solution Step 4 (Solve for E): $E = 12 + 0.6(5) = 15\,V$

Science of Electromagnetism

• Definition: Electromagnetism is the science of charge and the forces and fields associated with those charges. Electricity and magnetism are two aspects of this single phenomenon.

• Historical Development: They were long considered separate forces. In the 19th century, they were treated as interrelated. In 1905, Albert Einstein’s special theory of relativity established they are aspects of one common phenomenon.

• Differentiating Forces:     - Electric Forces: Produced by charges at rest or in motion. Responsible for physical and chemical properties of atoms. Enormously strong compared to gravity. Scaled example: Lightning/thunder.     - Magnetic Forces: Produced only by moving charges and act solely on charges in motion. Magnetism is due to forces between charges in motion.

• Field Interactions:     - Michael Faraday: A changing magnetic field produces an electric field (the basis for electric power generation).     - James Clerk Maxwell: A changing electric field produces a magnetic field.

Real-World Applications

• Incandescent Lightbulbs: Current heats a thin filament until it glows.

• Synchronization Systems: Electric clocks and traffic lights linked to vehicular flow.

• Communication: Radio and TV receive information via EM waves traveling at the speed of light.

• Automobiles: Starter motors use magnetic fields generated by current to rotate the shaft/engine pistons; the spark ignition is a momentary current discharge.

Electromagnetic Induction Principles

• Definition: The production of an electromotive force (EMF) across an electrical conductor in a changing magnetic field.

• Discovery: Michael Faraday (1831). Mathematically described by James Clerk Maxwell.

• Core Principle: A changing magnetic field induces a voltage. This is the foundation for generators and transformers.

Faraday’s Laws of Induction

• Faraday's First Law: Any change in the magnetic field of a coil of wire will cause an EMF to be induced in the coil.

• Faraday's Second Law: The magnitude of the induced EMF in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.

• Mathematical Formula: $\varepsilon = -N \times \frac{d\Phi}{dt}$     - $\varepsilon$ = Induced voltage/EMF (volts)     - $N$ = Number of turns in the coil     - $\Phi$ = Magnetic flux (Webers)     - $t$ = Time (seconds)

Lenz’s Law (1834)

• Statement: The direction of an induced current will always be such that it opposes the change or motion that produced it.

• Mathematical Representation: The negative sign in Faraday's law represents Lenz's Law.

Magnetic Flux Specifics

• Magnetic Flux ($\Phi$): Defined by the surface integral of the magnetic field over a region. $\Phi = \int \mathbf{B} \cdot d\mathbf{A}$

• Units:     - Magnetic Flux (Webers - $Wb$): One Weber is the flux linking a circuit if the induced EMF is $1\,V$ when the flux is reduced uniformly to zero in $1\,s$.     - Magnetic Field Strength ($B$): Measured in Teslas ($T$).

• Visualizing Flux: Proportional to the number of magnetic field lines passing through a loop.

• Methods to Vary Flux and Generate EMF:     1. The magnetic field $B$ changes (moving a magnet or using AC source).     2. The wire loop is deformed (surface area $\Sigma$ changes).     3. The orientation of the surface changes (spinning the loop in a fixed field).

Inductance: Self and Mutual

• Inductance Definition ($L$): The tendency of an electrical conductor to oppose a change in the electric current flowing through it. It acts as a form of electrical "inertia."

• The Henry ($H$): The SI unit of inductance. $1\,Henry$ is the inductance required to produce an EMF of $1\,volt$ when the current changes at the rate of $1\,Ampere\,per\,second$.

• Formula for Inductance: $L = \frac{\text{Induced Voltage}}{\frac{dI}{dt}}$

Self-Inductance

• Mechanism: A changing current in a coil produces a changing magnetic flux, which induces an EMF in the same coil that opposes the current change.

• Mathematical Relation: $\Phi \propto I \rightarrow \Phi = LI$ $\varepsilon = -L \frac{dI}{dt}$ $L = N \frac{\Phi}{I}$

Mutual Inductance

• Mechanism: A change in current in one coil induces an EMF in a second, separate nearby coil. The second coil opposes the change in the first.

Factors Affecting Inductance

• Number of Wire Turns ($N$): Inductance is greater with more turns; more turns create a greater magnetic field force for a given current.

• Coil Area: Inductance is directly proportional to coil area. Larger areas present less opposition to magnetic flux formation.

• Core Material: Inductance increases with the magnetic permeability of the core material around which the coil is wrapped.

• Coil Length: Inductance is inversely proportional to length. A longer coil has less inductance, while a shorter coil has greater inductance.

Review Questions and Practice Problems

1. State Faraday’s law of electromagnetic induction.

2. State Lenz’s law.

3. Explain the difference between self-inductance and mutual inductance.

4. Define electromotive force (EMF) and its SI unit.

5. Derive an expression for the EMF induced in a conductor moving in a magnetic field.

Practice Calculation:

• Problem: A $10\,\Omega$ resistor is connected to a battery of internal resistance $5\,\Omega$. The PD across the terminals of the battery is $2\,V$.     - (i) Find the current flowing: $I = \frac{V}{R} = \frac{2}{10} = 0.2\,A$     - (ii) Find the EMF of the battery: $E = I(R + r)$ $E = 0.2 \times (10 + 5) = 0.2 \times 15 = 3\,V$

Course Information and Learning Objectives

• Course Title: PHYSICS 121: General Physics III (Electromagnetic Induction)

• Credit Units: 2 Units

• Lecturer: Dr. Okafor P.N.

• Department: Department of Physics, College of Science and Technology, Covenant University

• Academic Session: 2024/2025

Student Learning Objectives

By the end of this lecture, students should be able to:

• Explain the term "electromotive force" (EMF).

• Describe high-level concepts of "electromagnetic induction."

• State the conditions necessary to increase or decrease induced EMF.

• Explain the concept of self-induction.

• Explain the concept of mutual induction.

Electromotive Force (EMF) and Potential Difference

• Definition of EMF: Electromotive force is a measure of electric potential difference that drives electrical current, expressed in volts ($V$). Despite its name, it does not refer to a mechanical force.

• Static Definition: EMF can be defined as the potential difference across the terminals of a cell when it is not delivering current (open circuit conditions).

• EMF vs. Terminal Voltage:

  • Electromotive Force (EMF): The potential difference of a source when no current is flowing ($I = 0$).

  • Terminal Voltage ($V$): The voltage output of a device measured across its terminals when part of a closed circuit.

• Mechanism of Flow: Electric potential difference generates an electric field, which exerts a force on charges, causing the flow of electric current.

• Energy Perspective: EMF represents the energy per unit electric charge imparted by an energy source (like a battery or generator). The work done on the electric charge transforms energy from one form (chemical or mechanical) to electrical, leading to one terminal becoming positively charged and the other negatively charged.

• Unit of EMF: The unit is the Volt ($V$). Driving one coulomb ($C$) of charge through a potential difference of one volt results in one joule ($J$) of work done.

• Work-Energy Formula: $ext{Electrical work in Joules} = ext{Charge} imes ext{Voltage}$

Circuit Equations and Internal Resistance

• The EMF Equation: The total EMF ($E$) of a cell accounts for both the external circuit and the internal resistance of the cell itself: $E = I imes (R + r)$

• Variables:

  • $E$ = Electromotive Force ($V$)

  • $I$ = Current flowing through the circuit ($A$)

  • $R$ = External resistance ($ext{Ω}$)

  • $r$ = Internal resistance of the cell ($ext{Ω}$)

• Terminal Potential Difference ($V$): This is the voltage across the external resistor: $V = I imes R$

• Voltage Drop due to Internal Resistance: $v = I imes r$

• Combined Relation: $E = V + (I imes r)$

Worked Examples: EMF and Resistance

• Example 1:

  • Problem: A circuit has an external resistor of $5 ext{Ω}$ and a cell with internal resistance of $0.5 ext{Ω}$. If the current flowing is $1 A$, calculate the EMF of the cell.

  • Solution: $E = I(R + r)$
    $E = 1 imes (5 + 0.5)$
    $E = 5.5 ext{ volts}$

• Example 2:

  • Problem: A cell with unknown EMF $E$ and internal resistance $2 ext{Ω}$ is connected to a $5 ext{Ω}$ resistor. The terminal potential difference ($V$) is measured as $1 V$. Calculate $E$.

  • Solution Step 1 (Calculate Current): Since Terminal PD ($V$) is across the $5 ext{Ω}$ resistor:
    $I = rac{V}{R} = rac{1}{5} = 0.2 A$

  • Solution Step 2 (Calculate Internal PD): PD across internal resistance $r$ is:
    $V_{internal} = I imes r = 0.2 imes 2 = 0.4 V$

  • Solution Step 3 (Calculate EMF):
    $E = V + V_{internal} = 1 + 0.4 = 1.4 V$

  • Alternative Solution:
    $E = I(R + r) = 0.2 imes (5 + 2) = 0.2 imes 7 = 1.4 V$

• Class Drill:

  • Problem: A wire of $4 ext{Ω}$ is connected to a battery of EMF $6 V$ and internal resistance $r$. The current is $1 A$. Calculate $r$.

  • Solution:
    $E = I(R + r) <br>ightarrow 6 = 1(4 + r)$
    $6 = 4 + r <br>ightarrow r = 2 ext{Ω}$

• Example 3:

  • Problem: The terminal PD of a battery is $12 V$ when an external resistance of $20 ext{Ω}$ is connected, and $13.5 V$ when an external resistance of $45 ext{Ω}$ is connected. Calculate the EMF ($E$) of the battery.

  • Solution Step 1 (Case 1): Current $I_1 = rac{12}{20} = 0.6 A$. Equation: $E = 12 + 0.6r$

  • Solution Step 2 (Case 2): Current $I_2 = rac{13.5}{45} = 0.3 A$. Equation: $E = 13.5 + 0.3r$

  • Solution Step 3 (Solve for r):
    $12 + 0.6r = 13.5 + 0.3r$
    $0.3r = 1.5 <br>ightarrow r = 5 ext{Ω}$

  • Solution Step 4 (Solve for E):
    $E = 12 + 0.6(5) = 15 V$

Science of Electromagnetism

• Definition: Electromagnetism is the science of charge and the forces and fields associated with those charges. Electricity and magnetism are two aspects of this phenomenon.

• Historical Development: Historically, electricity and magnetism were viewed as separate forces. In the 19th century, they were recognized as interrelated. In 1905, Einstein’s special theory of relativity established them as aspects of one common phenomenon.

• Differentiating Forces:

  • Electric Forces: Produced by charges at rest or in motion, responsible for the physical and chemical properties of atoms. They are significantly stronger than gravitational forces. Example: Lightning/thunder.

  • Magnetic Forces: Produced only by moving charges, acting solely on charges in motion. Magnetism derives from the interaction between charges in motion.

• Field Interactions:

  • Michael Faraday: A changing magnetic field induces an electric field (basis for electric power generation).

  • James Clerk Maxwell: A changing electric field produces a magnetic field.

Real-World Applications

• Incandescent Lightbulbs: Current heats a filament until it glows.

• Synchronization Systems: Electric clocks and traffic lights linked to vehicular flow.

• Communication: Radio and TV receive information via EM waves traveling at the speed of light.

• Automobiles: Starter motors utilize magnetic fields produced by current to operate; spark ignition involves momentary current discharge.

Electromagnetic Induction Principles

• Definition: The production of an electromotive force (EMF) across an electrical conductor in a varying magnetic field.

• Discovery: Made by Michael Faraday in 1831; mathematically described by James Clerk Maxwell.

• Core Principle: A changing magnetic field induces voltage, foundational for generators and transformers.

Faraday’s Laws of Induction

• Faraday's First Law: Any change in the magnetic field of a coil of wire will induce an EMF in the coil.

• Faraday's Second Law: The magnitude of induced EMF in a circuit is directly proportional to the rate of change of magnetic flux through the circuit.

• Mathematical Formula: egin{equation} ext{EMF } ( ext{ } ext{ } ext{ } extstyle ext{ extbf{ε}} ) = -N imes rac{d ext{Φ}}{dt} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }

  • $ext{ε}$ = Induced voltage/EMF (volts)

  • $N$ = Number of turns in the coil

  • $ext{Φ}$ = Magnetic flux (Webers)

  • $t$ = Time (seconds)

Lenz’s Law (1834)

• Statement: The direction of an induced current always opposes the change or motion that produced it.

• Mathematical Representation: The negative sign in Faraday's law represents Lenz's Law.

Magnetic Flux Specifics

• Magnetic Flux ($ext{Φ}$): Defined by the surface integral of the magnetic field over a region: $ext{Φ} = extstyle ext{ extbf{ } ext{ } } ext{ }$
  egin{equation} ext{Φ} = extstyle extbf{B} ext{ } extbf{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } extbf{ } ext{} ext{A} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } extbf{} ext{ } ext{$} ext{} ext{ } ext{} ext{ } ext{ } ext{ }

• Units:

  • Magnetic Flux (Webers - $Wb$): One Weber is the flux linking a circuit if the induced EMF is $1 V$ when the flux is uniformly reduced to zero in $1 s$.

  • Magnetic Field Strength ($B$): Measured in Teslas ($T$).

• Visualizing Flux: Proportional to the number of magnetic field lines passing through a loop.

• Methods to Vary Flux and Generate EMF:

  1. Change the magnetic field $B$ (moving a magnet or using AC source).

  2. Deform the wire loop (change the surface area $ext{Σ}$).

  3. Change the orientation of the surface (spin the loop in a fixed field).

Inductance: Self and Mutual

• Inductance Definition ($L$): The tendency of an electrical conductor to oppose a change in the electric current flowing through it, serving as a form of electrical "inertia."

• The Henry ($H$): The SI unit of inductance; $1 Henry$ is defined as the inductance required to produce an EMF of $1 volt$ when the current changes at the rate of $1 Ampere$ per second.

• Formula for Inductance: $L = rac{ ext{Induced Voltage}}{ rac{dI}{dt}}$

Self-Inductance

• Mechanism: A changing current in a coil generates changing magnetic flux, which induces an EMF in the same coil that opposes the current change.

• Mathematical Relation: ext{Φ} ext{ } ext{ } ext{ } m ext{ } extbf{ } ext{Flip this thing around to generate it } extbf{(byitalvol]}</$ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }
  $\bullet$ ext{ } $L$ = $N$ $extbf{}$ $c$ ≠_§ (-L(− ext {I}} $. <br>$L = N rac{ ext{Φ}}{I}$</p></li></ul><h6>Mutual Inductance</h6><ul><li><p><strong>Mechanism:</strong> A change in current in one coil induces an EMF in a second, separate nearby coil, which opposes the change in the first.</p></li></ul><h6>Factors Affecting Inductance</h6><ul><li><p><strong>Number of Wire Turns ($N$$): Inductance increases with more turns; more turns result in a greater magnetic field force for a given current.

• Coil Area: Inductance is directly proportional to coil area; larger areas have less opposition to magnetic flux formation.

• Core Material: Inductance rises with the magnetic permeability of the core material around the coil.

• Coil Length: Inductance is inversely proportional to length; a longer coil has less inductance, while a shorter coil has greater inductance.

Review Questions and Practice Problems

1. State Faraday’s law of electromagnetic induction.

2. State Lenz’s law.

3. Explain the difference between self-inductance and mutual inductance.

4. Define electromotive force (EMF) and its SI unit.

5. Derive an expression for