EEF 262: Physics for Engineering II Course Notes

General Course Information

• Course Code: EEF 262

• Title: Physics for Engineering II

• Credits: 4 Credits

• Semester Duration: 40-20-0 hours

• Instructor: Dr. DJOB Roger

• Institution: University of Buea

• Date: April 2026

Table of Contents

• List of Figures ii

• List of Tables iii

1. Magnetism 2
      - 1.1 Magnetism and Magnetic Field 2
        - 1.1.1 Definition of magnetism 2
        - 1.1.2 Definition of magnetic field 2
        - 1.1.3 What happens when two magnets are brought close together? 3
        - 1.1.4 Ferromagnetic Materials 3
      - 1.2 Electromagnetism 3
      - 1.3 Magnetic flux and flux Density 4
      - 1.4 Permeability 5
      - 1.5 Reluctance 6
      - 1.6 Ohm’s Law for Magnetic circuits 6
      - 1.7 Magnetic Field Intensity 7
      - 1.8 The Relationship between B and H 7
      - 1.9 Magnetic circuits 7
      - 1.10 Air Gaps, Fringing, and Laminated Cores 8

2. Ampere Theorem 10
      - 2.1 Definition 10
      - 2.2 Applications of the Ampere’s theorem 11
        - 2.2.1 Example 1: Calculating Line Integrals 11
        - 2.2.2 Example 2: Co-axial cable 12
        - 2.2.3 Example 3: Cylindrical Conductor 12
        - 2.2.4 Example 4: Two long solenoids 13

3. Biot and Savart Law 16
      - 3.1 Statement of the Biot and Savart law 16
      - 3.2 Determination of a magnetic field using Biot and Savart law 16
        - 3.2.1 Vector Notation 17
        - 3.2.2 Magnitude 17
      - 3.3 Applications of the Biot and Savart law 17
        - 3.3.1 B Field produced by a long, straight current conducting wire 17
        - 3.3.2 B Field on axis of circular current loop 19
        - 3.3.3 Field at Center of Partial Loop 19
        - 3.3.4 Partial Loops 20
        - 3.3.5 B Field produced by a solenoid 20

Chapter 1: Magnetism

1.1 Magnetism and Magnetic Field

1.1.1 Definition of Magnetism

• Magnetism is defined as the force of attraction or repulsion that acts between magnets and other magnetic materials.

• Common manifestations include:
    - Attraction of magnets to iron.
    - Deflection of compass needles.
    - Interaction between different magnets (attraction or repulsion).

1.1.2 Definition of Magnetic Field

• The magnetic field is defined as the region where the force exerted by a magnet is felt.

• It can be visualized using flux lines or lines of force, which illustrate the direction and intensity of the magnetic field at various points.

• Key characteristics include:
    - Strongest at the poles where flux lines are densest.
    - Direction from the north (N) to the south (S) outside the magnet.
    - Flux lines do not cross.

• The symbol for magnetic flux is Φ.

1.1.3 What Happens When Two Magnets Are Brought Close Together?

• Unlike poles attract each other: Flux lines pass between the two magnets.

• Like poles repel: The flux lines push back against each other, resulting in a flattened field between the magnets.

1.1.4 Ferromagnetic Materials

• Ferromagnetic materials include iron, nickel, cobalt, and their alloys, which are attracted by magnets.

• Such materials create an easier pathway for magnetic flux.

• Flux lines prefer to traverse ferromagnetic materials (such as soft iron) over nonmagnetic materials (like plastic or glass), which do not affect the magnetic field.

1.2 Electromagnetism

• Most applications of magnetism stem from magnetic effects due to electric currents.

• The current (I) in a conductor generates a concentric magnetic field around it.

• The magnetic field strength is directly proportional to the magnitude of the current.

• Right-Hand Rule:
    - The thumb points in the direction of the current, and the fingers curl in the direction of the magnetic field.

• The field direction and strength are affected by the coil's configuration, especially if it is wound around a ferromagnetic core.
    - If wound on a ferromagnetic core, most flux is confined within the core, resulting in a larger effective field.

1.3 Magnetic Flux and Flux Density

• Magnetic flux (Φ) represents the total magnetic field passing through a given area, measured in webers (Wb).

• Flux Density (B) is the magnetic flux per unit area, with units of $ext{Wb/m}^2$; this is called the tesla (T).

• Relation: $B = rac{ ext{Φ}}{A}$ where A is the area.

• Example Calculation:
    $ext{If at cross section 1, } B_1 = 0.4 ext{ T} ext{, To find } B_2:$
    - Using relation: $ext{Φ} = B_1 imes A_1 = B_2 imes A_2$
    - Therefore, $B_2 = B_1 imes rac{A_1}{A_2} = 0.4 imes rac{2 imes 10^{-2}}{1 imes 10^{-2}} = 0.8 ext{ T}$

1.4 Permeability

• Permeability (µ) is the measure of how easily magnetic flux lines can be established in a material.

• Free space permeability: $ext{µ}_0 = 4 ext{π} imes 10^{-7} ext{ Wb/A.m}$

• Materials with higher permeability allow more flux through; they can be categorized as follows:
    - Diamagnetic: Slightly less permeability than free space.
    - Paramagnetic: Slightly more permeability than free space.
    - Ferromagnetic: Significantly higher permeability, where $ext{µ}_r ext{ (relative permeability)} ext{ is typically } ext{µ}_r ≥ 100$.

1.5 Reluctance

• Reluctance (R) determines a material’s opposition to the establishment of a magnetic field, analogous to electrical resistance.

• Expression for reluctance:
    $R = rac{l}{ ext{µ} imes A} ext{ (At/Wb)}$
    Where,
    - l = length of the magnetic path.
    - A = cross-sectional area.

1.6 Ohm’s Law for Magnetic Circuits

• The relation between flux (Φ), magnetomotive force (MMF), and reluctance (R) is given by:
    $Φ = rac{F}{R} ext{(Wb)}$

• The MMF (F) is calculated as $F = NI ext{ (At)}$ where N is the number of turns, and I is the current:
    $Φ = rac{NI}{R}$

• Example:
    - For $R = 12 imes 10^{4} ext{ (At/Wb)}, ext{if } F = 300 imes 0.5 = 150 ext{ At},$
    - Therefore, $Φ = rac{150}{12 imes 10^{4}} = 1.25 imes 10^{-3} ext{ Wb}$

1.7 Magnetic Field Intensity

• Magnetic field intensity (H) is defined as the magnetomotive force per unit length of the path:
    $H = rac{F}{l} = rac{NI}{l} ext{ (At/m)}$

• The rearrangement provides a significant result, illustrating the relationship between MMF and magnetic field intensity:
    $NI = Hl$

1.8 The Relationship Between B and H

• The relationship can be described by:
    $B = ext{µ}H$

• Higher permeability (µ) correlates to higher flux density (B) for a consistent magnetizing current.

• In air, $B_g$ can be expressed in terms of $H_g$ as:
    $B_g = ext{µ}_0H_g; ext{where, } ext{µ}_0 = 4 ext{π} imes 10^{-7}$

1.9 Magnetic Circuits

• Magnetic circuits are networks that guide the magnetic flux, often seen in motors, generators, and transformers, influencing devices such as ATM cards through magnetic stripes.

• Magnetic circuits structure shapes the path of magnetic flux.

1.10 Air Gaps, Fringing, and Laminated Cores

• Fringing refers to the spreading of flux lines when there is an air gap, which reduces effective flux density in the gap.

• For small gaps, fringing can generally be neglected or accounted for by enlarging the cross-sectional area dimensions to compensate for losses.

Chapter 2: Ampere Theorem

2.1 Definition

• Ampere’s Law states that the integral around a closed path of the tangential component of the magnetic field (B) equals the permeability of free space times the total current enclosed by the path:
     ext{∮} B ullet ds = ext{µ}_0 I

2.2 Applications of the Ampere’s Theorem

2.2.1 Example 1: Calculating Line Integrals

• In a system with multiple conducting loops, calculate the net magnetic field based on currents.

• Using the right-hand rule to establish positive and negative current contributions:
    - ext{For Path 1: } ∮ B ullet dS = ext{µ}_0(i_2 - i_1)
    - ext{For Path 2: } ∮ B ullet dS = ext{−µ}_0(i_2 + 2i_1)

2.2.2 Example 2: Co-Axial Cable

• The enclosed current for varying radial distances within a coaxial cable is determined:
    - $I_{ ext{enc}} = I_0 rac{ ext{π}r^2}{ ext{π}a^2} = rac{I_0 r^2}{a^2}$
    - Application of Ampere's Law gives:
    - For radius r:
  $B = rac{ ext{µ}_0 I_0 r}{2 ext{π} a^2}$
  - For outer radius a, the magnetic field is:
    - $B = rac{ ext{µ}_0 I_0 }{2 ext{π}r}$
  - And for diffusing population of conductors, enclosed current decreases, yielding zero magnetic field at outer region.

2.2.3 Example 3: Cylindrical Conductor

• For a long cylindrical conductor with a non-uniform current density defined as $J = αr^2$, where $α$ is constant:
    - Magnetic field inside the conductor:
    - $I_{ ext{enc}} = ext{∫}_{0}^{r} J dA = ∫_0^R 2παr'^3 dr'$
    - On applying Ampere's law yields:
  $B = rac{µ_0 α r^3}{4}$
  - For outside the conductor or radius greater than R results in constant $B = rac{µ_0 α R^4}{4r}$

2.2.4 Example 4: Two Long Solenoids

• For two nested long solenoids with opposite currents, utilize Ampere's law to determine:
    - Each segment’s contributions focusing on net current enclosed.
    - For sections between and outside solenoids react in context of current density.

Chapter 3: Biot and Savart Law

3.1 Statement of the Biot and Savart Law

• The magnetic field (B) due to a steady current in a straight wire segment relates the field strength to the distance (r) from the wire and the current (i):
    $B ext{ is proportional to } i ext{ and inversely proportional to } r^2$

3.2 Determination of a Magnetic Field Using Biot and Savart Law

3.2.1 Vector Notation

• The differential magnetic field (dB) at any point P due to an infinitesimally small segment of wire is expressed as:
    $ext{dB} = rac{ ext{µ}_0 i}{4 ext{π}} rac{ ext{dS} imes ext{r}}{r^3}$

• The direction of dB follows the right-hand rule and forms a trihedron with vectors dS, r, and dB.

3.2.2 Magnitude

• The magnitude can be given by:
    $ext{dB} = rac{ ext{µ}_0 i}{4 ext{π}} rac{ ext{ds} ext{sin} θ}{r^2}$

3.3 Applications of the Biot and Savart Law

3.3.1 B Field Produced by a Long, Straight Current Conducting Wire

• The magnetic field produced at a distance R from a long straight current-carrying wire can be derived using the Biot-Savart Law, yielding:
    $B = rac{ ext{µ}_0 I }{2 ext{π}R}$

3.3.2 B Field on Axis of Circular Current Loop

• For a circular loop with N turns:
    - The field at the center is:
    $B = rac{N ext{µ}_0 i}{2R}$
    - Field on the axis:
    $B = rac{N ext{µ}_0 i}{2R^2} ext{ (for distance z off center)}$

3.3.3 Field at Center of Partial Loop

• For a circular partial loop covering angle Φ:
    - The field is given by:
    $B = rac{µ_0 i}{2R} rac{Φ}{2π}$
    - For half loops:
      $B = rac{µ_0 i}{4}( rac{1}{R1} - rac{1}{R2})$

3.3.4 Partial Loops

• For evaluating fields at points from several partial loops recognizing:
    - Only angles contribute to the field strength, and ensure directions must be respected for cumulative effect.

3.3.5 B Field Produced by a Solenoid

• A solenoid's field is derived with total current and number of turns per unit length, resulting in:
    - $dB = rac{ ext{µ}_0}{2} rac{NI}{L}$ along the central axis for practical use.

• For infinitely long solenoids:
    $B = rac{ ext{µ}_0 NI}{L}$

List of Figures

1. 1.1 Flux lines

2. 1.2 Attraction-repulsion phenomena

3. 1.3 Right hand rule

4. 1.4 Flux in a ferromagnetic core

5. 1.5 Application of magnetic circuits

6. 1.6 Air gaps

7. 2.1 Two conducting loops having currents

8. 2.2 Co-axial cable

9. 2.3 Two long solenoids nested on the same axis

10. 3.1 Schematic representation of a magnetic field produced by current distribution

11. 3.2 Schematic representation of a long, straight current conducting wire

12. 3.3 Schematic representation of a circular current loop with N turns

13. 3.4 Schematic representation of a circular partial loop with N turns

14. 3.5 Partial loops

15. 3.6 Solenoid representation.