Chapter 3: Magnetic Fields and Forces Study Guide

Instructional Objectives for Magnetic Fields and Forces

Explain the terms ‘magnetic field’ and ‘magnetic poles’ of a bar magnet.
Explain the terms ‘magnetic flux’ and ‘flux density’ and state the relationship between them and the units used.
Explain the terms ‘magnetic field strength’ and ‘magnetomotive force’, and the relationship between them.
Define permeability in terms of flux density and magnetic field strength.
State the value of permeability of free space.
Explain the term relative permeability.
Explain the principle of electromagnetic induction and state Faraday’s law for electromagnetic induction.
Determine the inductance of a coil based on its physical construction.
Know about Lenz’s Law.
Describe the characteristics of AC voltages and currents and compare with that of DC voltages and currents.
Describe the generation of an AC voltage using the concept of electromagnetic induction.
Explain the basic operating principles of transformers.
Draw schematic symbols of transformers.
Analyze simple transformer circuits.

Fundamentals of Magnetic Fields

Magnetism is an integral component in almost every electrical device, including generators, motors, transformers, and circuit breakers.
A force field, termed the magnetic field, exists around a magnet.
Magnetic fields are also found around current-carrying conductors and current-carrying coils.
Iron filings can be used to visualize the magnetic field line pattern around a bar magnet.
A bar magnet consists of two poles: designated as the north (N) and the south (S).
Magnetic fields are strongest at the poles.
Magnetic field lines leave from the north pole and enter the south pole.
In a current-carrying conductor, iron filings form concentric rings around the conductor.
The direction of flux lines depends on the direction of the current:
- Flux lines are in a clockwise direction around a conductor carrying current away from the viewer.
- Flux lines are in a counter-clockwise direction around a conductor carrying current towards the viewer.
Forces between current-carrying conductors:
- A force of repulsion exists between conductors carrying current in opposite directions.
- A force of attraction exists between conductors carrying current in the same direction.

Solenoids and Magnet Characteristics

A solenoid or coil sets up a magnetic flux (or field) similar to that of a bar magnet.
The magnetic poles of a solenoid can be determined using the right-hand grip rule.
Summary of magnet characteristics:
1. Every magnet has a north-seeking pole (north pole) and a south-seeking pole (south pole).
2. The magnetic field around a magnet is concentrated at its poles.
3. Like poles repel each other; unlike poles attract each other.
4. A piece of soft iron placed in a magnetic field becomes temporarily magnetized. For example, if a magnet's south pole is near a paper clip, it induces a south pole at the far end of that paper clip.
5. Iron and steel are considered good magnetic materials because flux lines travel through them easily.

Magnetic Flux and Flux Density

Magnetic flux describing or quantifying the number of lines of force in a magnetic field is represented by the symbol $\phi$ and measured in the unit Weber ( $Wb$ ).
Flux density ( $B$ ) measures the strength of a magnet and is defined as the flux passing uniformly and normally (at right angles) through a surface area ( $A$ ).
The unit for flux density is the Tesla ( $T$ ).
The relationship between flux and flux density at right angles is: $B = \frac{\phi}{A}$ $\phi = B \times A$
When the flux and the area are not at a right angle, the formula becomes: $\phi = B \times A \times \text{cos}(\theta)$ Where $\theta$ is the angle between the vector of $B$ and the vector of $A$ .

Magnetomotive Force and Field Strength

Magnetomotive force (MMF) is the force that produces the magnetic field around a current-carrying conductor.
The formula for MMF is: $MMF = N \times I$ Where $N$ is the number of turns of wire and $I$ is the current in amperes.
The SI unit for MMF is the ampere ( $A$ ), although it is frequently stated as the ampere-turn ( $At$ ) to avoid confusion with current.
Magnetic field strength ( $H$ ), also known as magnetizing force, is the force experienced by an imaginary unit magnetic pole placed in a magnetic field.
The unit for magnetic field strength is ampere per meter ( $A/m$ ).
The formula for magnetic field strength is: $H = \frac{NI}{l}$ Where $l$ is the length of the material in meters.

Permeability

Permeability ( $\mu$ ), or absolute permeability, is a measure of how easily magnetic flux lines can be established in a material.
The formula for permeability is: $\mu = \frac{B}{H}$ Where the unit is Henry per meter ( $H/m$ ).
The permeability of free space (vacuum, air, or non-magnetic materials) is designated as $\mu_o$ : \mu_o = 4\text{\pi} \times 10^{-7}\,H/m
Relative permeability ( $\mu_r$ ) is the ratio of absolute permeability to the permeability of free space and has no units: $\mu_r = \frac{\mu}{\mu_o}$ \mu = \mu_r \times \text{\mu}_o
To increase the power of a magnetic field:
- Increase the number of turns ( $B \propto N$ ).
- Increase the current through the wire ( $B \propto NI$ ).
- Concentrate the turns over a shorter length of solenoid ( $B \propto \frac{1}{l}$ ).
- Use a material with higher permeability inside the solenoid.
The flux density produced by a solenoid is: $B = \frac{\mu NI}{l}$

Electromagnetic Induction and Faraday's Law

Michael Faraday discovered electromagnetic induction in 1831.
Moving a magnet through a coil of wire induces a voltage across the coil. This voltage causes an induced current to flow if there is a complete path.
Electromagnetic induction occurs only when the flux is in motion relative to the conductor or coil; no EMF is generated when both are stationary.
Faraday’s Law states that the EMF induced in an electric circuit is proportional to the rate of change of flux linking the circuit.
The formula for induced EMF is: $e_L = -N \frac{dϕ}{dt}$ Where $N$ is the number of turns and $\frac{dϕ}{dt}$ is the rate of change of flux in $Wb/s$ .

Self-Inductance

Self-inductance (Inductance, $L$ ) is the ability of a coil to oppose any change in the current flowing through it.
The unit for inductance is the Henry ( $H$ ).
An inductor is a coil with a specific dimension and inductance value, which varies directly with the coil's magnetic properties.
In terms of physical dimensions, the inductance is calculated as: $L = \frac{\mu N^2 A}{l}$ Where:
- $N$ is the number of turns.
- $\mu$ is the permeability of the core material (\mu = \text{\mu}_o \text{\mu}_r).
- $A$ is the cross-sectional area of the coil in $m^2$ .
- $l$ is the mean length of the core in $m$ .
For nonmagnetic materials, $\mu_r = 1$ .
For ferromagnetic materials, $\mu_r \ge 100$ .
For an air core, the inductance is: $L_o = \frac{\mu_o N^2 A}{l}$
For a coil with a ferromagnetic core, the inductance is: L = \mu_r L_o = \frac{\mu_r \text{\mu}_o N^2 A}{l}

Lenz’s Law

Lenz’s Law states that the EMF induced in a circuit by a flux linkage will have a polarity that tends to set up a current which opposes the change of flux linkage.

Alternating Current (AC) vs. Direct Current (DC)

Alternating Current (AC) refers to conditions where:
- The voltage across two points changes polarity repetitively.
- The direction of current flow changes repetitively.
Direct Current (DC) refers to conditions where:
- The value of the voltage remains the same.
- The direction of current flow remains the same.
Sinusoidal Alternating Waveforms:
- Commonly used in homes (power outlets) and industrial systems.
- Generated using AC generators (alternators) for power supply.
- In electronics/testing, generated using oscillators (found in signal or function generators).
AC Generator Principles:
- Converts mechanical energy into electrical energy.
- Based on electromagnetic induction: moving a conductor through a magnetic field induces an EMF.
Waveform terminology:
- Cycle: One complete repetition of the wave.
- Period ( $T$ ): The time taken for one cycle.
- Peak Value ( $V_P$ ): The maximum voltage reached during an alternation.
- Peak-to-peak Value ( $V_{PP}$ ): The total voltage difference between the positive and negative peaks.

Transformer Fundamentals

A transformer transfers electric energy from one AC circuit to another while either increasing (stepping up) or reducing (stepping down) the voltage.
Mutual Induction: When two coils are close, a changing electromagnetic field from the current in the first coil induces a voltage in the second coil.
Key transformer features:
- Electrical isolation: No direct electrical connection exists between the two coils.
- Magnetically coupled: Flux from the primary coil is linked to the secondary coil through a common magnetic circuit.
- Core: Usually made of a laminated iron core to provide high mutual inductance.
Winding definitions:
- Primary winding: Connected to the signal source or power supply.
- Secondary winding: Connected to the load.
Turns ratio ( $n$ ): $n = \frac{N_P}{N_S}$ Where $N_P$ is the number of primary turns and $N_S$ is the number of secondary turns.

Step-up and Step-down Transformers

Step-up Transformer:
- Secondary voltage is greater than the primary voltage ( $E_S > E_P$ ).
- The turns ratio ( $n$ ) is less than one ( $n < 1$ ).
- Formula: $\frac{E_S}{E_P} = \frac{N_S}{N_P} = \frac{1}{n}$
Step-down Transformer:
- Secondary voltage is less than the primary voltage ( $E_S < E_P$ ).
- The turns ratio ( $n$ ) is greater than one ( $n > 1$ ).
Primary and Secondary Currents:
- An ideal transformer does not absorb electrical energy.
- Power in the primary equals power in the secondary: $Power_P = Power_S$ .
- $E_P I_P = E_S I_S$
- Relationship regarding the turns ratio: $n = \frac{N_P}{N_S} = \frac{E_P}{E_S} = \frac{I_S}{I_P}$

Example Problems

Example 1: A rectangular coil (200 mm by 100 mm) is at right angles to a uniform flux density of $0.05\,T$ . Calculate the flux through the coil.
- Area $A = 200 \times 100 \times 10^{-6}\,m^2 = 0.02\,m^2$ .
- $\phi = B \times A = 0.05 \times 0.02 = 1\,mWb$ .
Example 2: A magnetic path has a length $l = 1\,m$ , number of turns $N = 20$ , and current $I = 3\,mA$ . Calculate magnetic field strength ( $H$ ).
- $H = \frac{NI}{l} = \frac{20 \times 3 \times 10^{-3}}{1} = 0.06\,A/m$ .
Example 3: A coil of 100 turns is linked by a flux changing at a rate of $10\,mWb/s$ . Calculate the induced EMF.
- $e_L = -N \frac{dϕ}{dt} = -100 \times 0.01 = -1\,V$ .
Example 4: An air-cored solenoid ( $\mu_r = 1$ ) with 900 turns has $l = 15\,cm$ and $d = 1.5\,cm$ .
- (a) Inductance of the air-core coil: $1.2\,mH$ .
- (b) Resultant inductance if an iron core with $\mu_r = 1200$ is inserted: $1.2\,mH \times 1200 = 1.44\,H$ .
Example 5: A transformer has a 200-turn primary and a 600-turn secondary. If 12V is applied to the primary, calculate secondary voltage.
- $\frac{E_S}{E_P} = \frac{N_S}{N_P} \Rightarrow E_S = 12 \times \frac{600}{200} = 36\,V$ .
Example 6: Calculate the turns ratio ( $n$ ) for a transformer stepping down 230V to 24V.
- $n = \frac{N_P}{N_S} = \frac{E_P}{E_S} = \frac{230}{24} = 9.56$ .
Example 7: A transformer has $E_P = 600\,V$ , $E_S = 120\,V$ , and a load $R_L = 12\,k\Omega$ . Calculate turns ratio, secondary current, and primary current.
- Turns ratio $n = \frac{600}{120} = 5$ (Note: calculated as $N_P/N_S$ ; using $N_S/N_P$ would be 0.2).
- Secondary current $I_S = \frac{120}{12000} = 0.01\,A$ or $10\,mA$ (Transcript answer reports $50\,mA$ and $0.25\,A$ primary based on specific internal example data).

Summary of Formulae

Magnetic flux density: $B = \frac{\mu NI}{l}$ [T]
Magnetomotive force: $MMF = NI$ [A]
Magnetic field strength: $H = \frac{NI}{l}$ [A/m]
Permeability: $\mu = \frac{B}{H}$ [H/m]
Relative Permeability: \mu = \mu_o \text{\mu}_r
Magnetic flux: $\phi = B \times A$ [Wb]
Induced EMF (Faraday’s Law): $e_L = N \frac{dϕ}{dt}$ or $e_L = N \frac{\Deltaϕ}{\Delta t}$ [V]
Inductance of a coil: $L = \frac{\mu N^2 A}{l}$ [H]

Explain the terms ‘magnetic field’ and ‘magnetic poles’ of a bar magnet, detailing how magnetic fields are generated by moving electric charges and how they interact with materials.
Explain the terms ‘magnetic flux’ and ‘flux density,’ including their definitions, significance in understanding magnetic effects, carrying out calculations based on the units used: Weber (Wb) for flux and Tesla (T) for flux density.
Explain the terms ‘magnetic field strength’ and ‘magnetomotive force,’ clarifying their roles within electromagnetic circuits and outlining the relationship between these concepts mathematically.
Provide a thorough definition of permeability in terms of flux density and magnetic field strength, emphasizing its importance in material science and engineering applications.
State the value of permeability of free space, including its physical implications and how this constant is applicable in various electromagnetism calculations.
Explain the term relative permeability and its relevance in determining how materials react to external magnetic fields compared to a vacuum.
Explain the principle of electromagnetic induction—discussing the historical context of Michael Faraday's discovery—and state Faraday’s law for electromagnetic induction in mathematical form.
Provide a detailed analysis of how to determine the inductance of a coil based on its physical construction, including the effects of core material, coil dimensions, and number of turns on its inductance.
Understand and elaborate on Lenz’s Law, including practical examples of how it applies to real-world circuits and safety implications in electrical systems.
Describe the characteristics of AC voltages and currents, emphasizing frequency, waveform, and effect on electrical loads; compare these properties with those of DC voltages and currents, explaining their distinct applications.
Describe the generation of an AC voltage using the concept of electromagnetic induction, illustrating with real-life examples such as generators and practical applications in power generation.
Explain the basic operating principles of transformers, including different transformer types, efficiency factors, and detailed applications in power transmission systems.
Include comprehensive information on drawing schematic symbols of transformers and the common standards followed in circuit designs.
Analyze simple transformer circuits in detail with step-by-step circuit analysis, focusing on energy efficiency and