# Fundamentals of Magnetic Flux and Reluctance

Understanding the fundamentals of electric flux and reluctance is crucial to solving transformer flux and magnetic circuits. As per the guidelines set by the NCEES®, the first part of this guide will help you uncover the fundamental concepts pivotal in understanding electric flux and reluctance.

Let’s dive deep into the details.

## Understanding Magnetic Flux

Magnetic flux is a fundamental concept in electromagnetism that describes the quantity of magnetic field passing through a surface or area.

In simpler terms, it measures the strength of a magnetic field passing through a given surface. Magnetic fields are produced by magnetic sources such as magnets or electric currents flowing through conductors.

### Magnetic Lines of Force

Magnetic lines of force, also known as magnetic field lines or magnetic flux lines, are imaginary lines used to represent the direction and strength of a magnetic field in space.

They visually represent how magnetic field lines interact with a magnetic source and other magnetic materials. Magnetic lines of force are always continuous loops that form closed paths, never intersecting each other.

The critical properties of magnetic lines of force are as follows:

- They emerge from the magnet’s north pole and enter the south pole, creating a closed loop within the magnet.
- The direction of the magnetic field at any point along the magnetic field line is tangential to the line at that point.
- The density of the magnetic field lines indicates the strength of the magnetic field: the closer the lines are to each other, the stronger the magnetic field in that region.

#### Movement of Magnetic Lines of Force

Magnetic lines of force move from the north pole to the magnet’s south pole. Outside the magnet, they extend from the north pole and loop back to the south pole, forming a closed path. The magnetic field lines also loop inside the magnet from the north to the south pole, but this loop is confined within the magnet.

#### Strength of Magnetic Field Along the Magnetic Lines of Force

Moving Towards the Magnetic Source: As we move closer to a magnetic source (a magnet or a current-carrying wire), the density of magnetic field lines increases, indicating a stronger magnetic field. The magnetic field lines are more concentrated, suggesting that the magnetic flux is higher near the source.

Moving Away from the Magnetic Source: Conversely, as we move away from the magnetic source, the density of magnetic field lines decreases, indicating a weaker magnetic field. The magnetic field lines are more spread out, suggesting that the magnetic flux is lower and farther away from the source.

The strength of the magnetic field along the magnetic lines of force follows an inverse square law, similar to the intensity of light from a point source. As the distance from the source increases, the magnetic field strength diminishes rapidly.

### Magnetic Field and Magnetic Field Intensity

A magnetic field is a region in space where a magnetic force is experienced by magnetic materials or moving charges. The magnetic field intensity, denoted by H, represents the measure of the magnetizing force experienced by a magnetic material in the presence of the magnetic field.

It is related to the magnetic field strength B (also known as magnetic flux density) through the permeability of the material (μ) as follows:

B = μ * H,

Where:

- B is the magnetic field strength (measured in Tesla, T).
- H is the magnetic field intensity (measured in Ampere per meter, A/m).
- μ is the material’s permeability (measured in Henry per meter, H/m).

### Magnetic Field Area

The magnetic flux is measured through an area perpendicular to the magnetic field lines. This area is often referred to as the flux or cross-sectional area. The angle between the magnetic field lines and the normal vector (perpendicular) to the area is denoted by θ.

It is important to note that only the magnetic field component perpendicular to the area contributes to the magnetic flux.

### Permeability (μ) and Magnetic Field Around a Conductor

Permeability (μ) is a fundamental property of a material that describes how easily it allows magnetic flux lines to pass through it when subjected to a magnetic field.

In other words, it measures the material’s ability to become magnetized and sustain a magnetic field. Permeability is a crucial parameter in electromagnetism and is represented by the “μ.”

#### Types of Permeability

There are two types of permeability:

**Permeability of Free Space (μ0) –**This is the permeability of a vacuum or free space. It is a universal constant denoted by “μ0” (mu naught). The value of μ0 is approximately 4π × 10^-7 T·m/A (Tesla-meter per Ampere).**Permeability of a Material (μr) –**This is a specific material’s permeability relative to free space’s permeability. It is denoted by “μr” (mu r). The value of μr can be greater than, equal to, or less than 1, depending on the material.

#### Permeability in Biot-Savart’s Law

Biot-Savart’s law is a fundamental law in electromagnetism that describes the magnetic field generated by a steady current in a conductor.

The law states that the magnetic field (dB) at a point due to an infinitesimally small length element (dL) of a conductor carrying a steady current (I) is directly proportional to the product of the current, the length element, and the sine of the angle between the length element and the line connecting the length element to the point.

The formula for Biot-Savart’s law is given as follows:

dB = (μ0 / 4π) * (I * dL x r) / r^3,

Where:

- dB is the magnetic field at a point (measured in Tesla, T).
- μ0 is the permeability of free space (measured in T·m/A).
- I is the current flowing through the conductor (measured in Amperes, A).
- dL is the infinitesimal length element of the conductor (measured in meters, m).
- r is the distance between the length element and the point where the magnetic field is measured (measured in meters, m).

The significance of permeability (μ) in Biot-Savart’s law is the term (μ0 / 4π). This term quantifies how the surrounding medium affects the magnetic field strength. For a vacuum or free space, the permeability (μ0) is a constant, and the term reduces to (1 / 4π), which is approximately 10^-7 T·m/A. However, the permeability may vary for different materials, leading to different magnetic field strengths.

For materials with a permeability greater than μ0 (μr > 1), the magnetic field strength will be amplified compared to that in free space.

These materials are referred to as paramagnetic. Conversely, the magnetic field strength will be weakened for materials with a permeability less than μ0 (μr < 1). These materials are called diamagnetic.

### Mathematical Expression for Magnetic Flux

The mathematical expression for magnetic flux (Φ) passing through a surface area A is given by:

Φ = B * A * cos(θ)

Where:

- Φ is the magnetic flux (measured in Weber, Wb).
- B is the magnetic field strength (measured in Tesla, T).
- A is the area of the surface (measured in square meters, m²).
- θ is the angle between the magnetic field lines and the normal vector to the area.

### Unit of Magnetic Flux

To derive the unit of magnetic flux, we need to analyze the formula for magnetic flux and examine the units of its constituent components.

The magnetic flux (Φ) passing through a surface is given by the formula:

Φ = B * A * cos(θ)

Since:

- B is the magnetic field strength (measured in Tesla, T).
- A is the area of the surface (measured in square meters, m²).
- θ is the angle between the magnetic field lines and the normal vector to the area (dimensionless).

Now, let’s examine the units of each component:

**Magnetic field strength (B):**

B is measured in Tesla (T).

1 Tesla (T) = 1 Newton per Ampere-meter (N/A·m).

The unit of magnetic field strength, Tesla, represents the force experienced by a one-meter length of conductor carrying one Ampere of current, when placed perpendicular to the magnetic field.

**Area (A):**

A is measured in square meters (m²).

The unit of area is derived from the fundamental unit of length, meter (m), squared.

**Angle (θ):**

Angle θ is dimensionless and measured in degrees or radians.

Now, let’s put all the units together to derive the unit of magnetic flux:

Φ = B * A * cos(θ)

Unit of Φ = (Unit of B) * (Unit of A) * (Unit of cos(θ))

Unit of Φ = (N/A·m) * (m²) * (dimensionless)

Unit of Φ = N·m / A

The unit of magnetic flux is Newton-meter per Ampere (N·m/A). This derived unit is commonly known as Weber (Wb), named after the German physicist Wilhelm Eduard Weber, and it is the standard unit used to measure magnetic flux.

### Dot Product and Its Significance in Magnetic Flux

The dot product is a mathematical operation used to determine the component of one vector that acts in the direction of another vector.

In the context of magnetic flux, the dot product between the magnetic field vector (B) and the area vector (A) gives the component of B perpendicular to the area A.

Mathematically, the dot product is defined as follows:

For two vectors A and B in 3D space,

A · B = |A| * |B| * cos(θ)

Where:

- A · B is the dot product of A and B.
- |A| and |B| are the magnitudes (or lengths) of vectors A and B, respectively.
- θ is the angle between vectors A and B.

#### Cases of Maximum and Minimum Flux

- When
**θ = 0° (cos(0°) = 1)**, the magnetic field lines are perpendicular to the area vector. This results in the maximum magnetic flux passing through the area. - When
**θ = 90° (cos(90°) = 0)**, the magnetic field lines parallel the area vector. In this case, the magnetic flux passing through the area is zero (minimum), as no magnetic field lines cross the surface.

### Magnetic Flux vs. Electric Flux

Electric flux is a concept in electrostatics, similar in principle to magnetic flux. It measures the quantity of electric field passing through a surface. The electric flux (Φe) through an area A in the presence of an electric field E is given by:

Φe = E * A * cos(θ)

The expressions for magnetic and electric flux are remarkably similar, involving the dot product of the respective field and area vectors. However, they represent different physical phenomena—one deals with magnetic fields, and the other deals with electric fields.

For a detailed analysis and relationship between electric and magnetic fields, read our detailed study guide on Electromagnetics in the FE Electrical exam.

### Flux Linkage

Flux linkage is another important concept, especially in electrical circuits and electromagnetic induction. It measures the total magnetic flux that links with a specific coil or circuit.

#### Mathematical Expression for Flux Linkage

For a coil with N turns and magnetic flux Φ passing through each turn, the total flux linkage (Λ) is given by:

Λ = N * Φ

Where:

- Λ is the total flux linkage (measured in Weber-turns, Wb-turns).
- N is the number of turns in the coil.
- Φ is the magnetic flux passing through each turn.

Flux linkage is a crucial parameter in transformers, motors, and generators, as it plays a significant role in determining induced voltages and electromagnetic forces.

### Magnetic Flux Problem 1 – Calculating Magnetic Flux due to a Point Charge

Consider, we are given:

Charge of the point charge, q = 2 × 10^-6 C

The velocity of the charge, v = 5 m/s

Angle concerning the magnetic field, θ = 30°

Magnetic field strength, B = 0.1 T

Area of Surface, A = 0

Since, Magnetic flux, Φ = B * A * cos(θ) ≈ 0.1 * 0 * cos(30°) = 0

The magnetic flux through the point charge is Φ = 0 Wb. It means the Magnetic Flux can be minimum for B=0 or A=0 irrespective of the angle.

### Magnetic Flux Problem 2 – Calculating Magnetic Flux due to a Straight Wire Passing Through Hypothetical Loop

Suppose we have:

Current through the wire, I = 5 A

Distance from the wire, r = 0.2 m

Width of the hypothetical rectangular loop, w = 0.1 m

Height of the hypothetical rectangular loop, h = 0.2 m

Permeability of free space, μ0 ≈ 4π × 10^-7 T·m/A

Let’s find the magnetic flux around the wire.

**Step 1**

Magnetic field strength, B = (μ0 * I) / (2π * r)

So B = (4π × 10^-7 T·m/A * 5 A) / (2π * 0.2 m) = 5 × 10^-6 T

**Step 2**

Area, A = w * h

So A = 0.1 m * 0.2 m = 0.02 m²

**Step 3**

Angle, θ = 90°

(since the loop is perpendicular to the wire. That is; parallel to the magnetic field)

**Step 4**

Magnetic flux, Φ = B * A * cos(θ) = (5 × 10^-6 T) * (0.02 m²) * cos(90°) = 5 × 10^-7 Wb

The magnetic flux around the wire is Φ = 5 × 10^-7 Wb.

Magnetic Flux Problem 3: Calculating Magnetic Flux due to a Circular Loop

Consider the following data for a coil with a single turn.

Current through the circular loop, I = 2 A

The radius of the circular loop, R = 0.3 m

The radius of the second loop, r = 0.2 m

Permeability of free space, μ0 ≈ 4π × 10^-7 T·m/A

Let’s calculate magnetic flux:

**Step 1: Magnetic field strength, B = (μ0 * I) / (2 * R)**

B = (4π × 10^-7 T·m/A * 2 A) / (2 * 0.3 m) = 4.2 × 10^-6 T

**Step 2: Area, A = π * r^2**

A = π * (0.2 m)^2 ≈ 0.1257 m²

**Step 3: Angle, θ = 0° (since the loops are concentric)**

**Step 4: Magnetic flux, Φ = B * A * cos(θ) = (4.2 × 10^-6 T) * (0.1257 m²) * cos(0°) ≈ 5.28 × 10^-7 Wb**

The magnetic flux through the second circular loop is Φ ≈ 5.28 × 10^-7 Wb.

## Understanding Reluctance (R)

Reluctance is a concept in electromagnetism similar to the resistance in electric circuits. It quantifies the opposition of a magnetic circuit to the flow of magnetic flux. In other words, reluctance measures how difficult it is for magnetic flux lines to pass through a magnetic circuit when subjected to a magnetic field.

The term “reluctance” is derived from “reluctant,” meaning “unwilling,” suggesting that the material or magnetic circuit is unwilling to allow the magnetic flux to pass through easily.

### Mathematical Expression of Reluctance

The mathematical expression for reluctance (S) is given by the ratio of the magnetomotive force (MMF) to the magnetic flux in a magnetic circuit:

Reluctance (S) = MMF / Φ

Where:

- S is the reluctance of the magnetic circuit (measured in Ampere-turns per Weber, A-turn/Wb).
- MMF is the magnetomotive force (measured in Ampere-turns, A-turn).
- Φ is the magnetic flux passing through the magnetic circuit (measured in Weber, Wb).

The magnetomotive force (MMF) is similar to electromotive force (EMF) in electrical circuits and represents the “driving force” that causes magnetic flux to flow in the magnetic circuit.

### Derivation of the Unit “Reluctance” (A-turn/Wb)

As reluctance (S) is defined as the ratio of the magnetomotive force (MMF) to the magnetic flux (Φ) in a magnetic circuit:

Reluctance (S) = MMF / Φ

The unit of MMF is Ampere-turns (A-turn), and the unit of magnetic flux is Weber (Wb).

Thus, the unit of reluctance (S) can be derived as:

Unit of Reluctance (S) = (Unit of MMF) / (Unit of Φ)

The unit of MMF is Ampere-turns (A-turn), and the unit of magnetic flux (Φ) is Weber (Wb).

Unit of Reluctance (S) = A-turn / Wb

Therefore, the unit of reluctance is Ampere-turn per Weber (A-turn/Wb), which is also represented as H^-1 (Henry-inverse) in SI units.

### Relationship Between Reluctance, Magnetic Field, and Magnetic Permeance

The reluctance of a magnetic circuit is inversely proportional to the magnetic field strength (H) and directly proportional to the length (l), and inversely proportional to the permeability (μ) of the material of the magnetic circuit.

Reluctance (S) = l / (μ * A)

Where:

l is the length of the magnetic path (measured in meters, m).

A is the cross-sectional area of the magnetic path (measured in square meters, m²).

μ is the material’s permeability (measured in Henry per meter, H/m).

The reciprocal of reluctance is called magnetic permeance (P):

Magnetic Permeance (P) = 1 / S = (μ * A) / l

Simply:

- Magnetic permanence is just like conductance in electricity.
- Reluctance is just like resistance in electricity.

### Significance of Reluctance

Reluctance is a critical parameter in designing and analyzing magnetic circuits, such as magnetic cores in transformers, inductors, and other electromagnetic devices. It helps engineers optimize the design to ensure efficient magnetic flux transmission while minimizing losses due to magnetic resistance.

Consider a magnetic circuit with a length of 0.2 meters and a cross-sectional area of 0.001 square meters. The magnetic material used in the circuit has a permeability of 1000 H/m. Let’s calculate the reluctance of the magnetic circuit.

We have:

Length of magnetic circuit (l) = 0.2 m

Cross-sectional area (A) = 0.001 m²

Permeability of the material (μ) = 1000 H/m

**To calculate the reluctance (S) using the formula**

Reluctance (S) = l / (μ * A)

Substitute the given values for a given magnetic circuit example:

S = 0.2 m / (1000 H/m * 0.001 m²) = 0.2 m / 1 H = 0.2 A-turn/Wb

The reluctance of the magnetic circuit is 0.2 A-turn/Wb.

### Conclusion

Now you have a rich idea about electric flux and how it relates to different magnetic and electrical quantities. Magnetic flux problems and other magneto reluctance concepts are critical for electrical and power engineering students.

Understanding transformer flux calculations and magnetic equivalent circuits helps engineers analyze the behavior of transformers, predict their performance, and ensure optimal energy transfer between circuits.

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