Physics (9630) | Magnetic flux and flux linkage

Magnetic Flux and Flux Linkage: Capturing the Field

Welcome to a crucial topic in magnetic fields! In previous sections, we learned about the existence of magnetic fields and how they exert forces (Magnetic Flux Density, B). Now, we need a way to quantify how much of that magnetic field *passes through* a specific area, like a coil of wire.

This concept, called Magnetic Flux, is the foundation for understanding how generators, transformers, and many other crucial technologies work (a topic called Electromagnetic Induction, which you will study next). Don't worry if this seems abstract—we'll break it down using simple visuals!

1. Setting the Stage: Magnetic Flux Density (B) Review

Before diving into flux, let's quickly remind ourselves about Magnetic Flux Density (B).

• B measures the strength of the magnetic field.
• It is a vector quantity (it has direction).
• Its unit is the Tesla (T).

Imagine B as the intensity or flow rate of magnetic "rain" falling in a specific direction.

2. Magnetic Flux ($\Phi$): The Capture Area

Magnetic Flux ($\Phi$) tells you the total amount of magnetic field lines passing through a particular surface area.

Think of it like this: If B is the rain intensity, the magnetic flux ($\Phi$) is the amount of rain you collect with a bucket or through a window. The total amount collected depends not only on the rain intensity (B) but also on the size of your window (A) and how you hold it.

Definition and Formula (Perpendicular Case)

Magnetic flux ($\Phi$) is formally defined as the product of the magnetic flux density ($B$) and the area ($A$) perpendicular to the field lines.

The simplest formula applies when the area vector is aligned perfectly with the field (meaning the surface is perpendicular to the field lines):

$$\Phi = BA$$

• \(\Phi\) is the Magnetic Flux.
• \(B\) is the Magnetic Flux Density (T).
• \(A\) is the Area through which the flux passes (\(\text{m}^2\)).

Units of Magnetic Flux

The SI unit for magnetic flux ($\Phi$) is the Weber (Wb).

Since \(\Phi = BA\):

$$1 \text{ Wb} = 1 \text{ T m}^2$$

Key Takeaway: Magnetic flux measures the total field "coverage" over an area. The unit is the Weber (Wb).

3. The Role of Angle: Tilting the Area

What happens if the area is tilted relative to the magnetic field lines? This is where the angle becomes crucial.

To deal with angles, we use a concept called the Normal. The normal is an imaginary line drawn perpendicular (at 90°) to the surface area.

The General Formula for Magnetic Flux

When the magnetic field $B$ is uniform but not perpendicular to the area $A$, the magnetic flux is given by:

$$\Phi = BA \cos \theta$$

Where \(\theta\) is the angle between the Magnetic Flux Density vector ($B$) and the Normal to the Area ($A$).

Understanding the Angle ($\theta$)

The $\cos \theta$ factor ensures we only account for the component of the area that is perpendicular to the field (or the component of $B$ that is perpendicular to the surface).

Let's look at the three critical scenarios:

1. Maximum Flux ($\theta = 0^\circ$):
   

   Scenario: The surface is held perfectly perpendicular to the field lines. The Normal is parallel to the field lines.

   Math: If $\theta = 0^\circ$, then $\cos 0^\circ = 1$.

   Result: $\Phi = BA$. (Maximum flux collected.)

   Analogy: Holding the window perfectly facing the rain.

2. Zero Flux ($\theta = 90^\circ$):
   

   Scenario: The surface is held parallel to the field lines. The Normal is perpendicular to the field lines.

   Math: If $\theta = 90^\circ$, then $\cos 90^\circ = 0$.

   Result: $\Phi = 0$. (No flux collected.)

   Analogy: Holding the window sideways, so the rain passes along the plane of the window without entering.

3. Intermediate Flux ($0^\circ < \theta < 90^\circ$):
   

   The flux is some fraction of the maximum value.

💡 Memory Aid: Define Your $\theta$!

Struggling students often confuse the angle between the field and the surface ($\alpha$) with the angle between the field and the normal ($\theta$). ALWAYS use the angle with the NORMAL.

If a question gives you the angle the surface makes with the field ($\alpha$), remember: $\theta = 90^\circ - \alpha$.

4. Magnetic Flux Linkage ($N\Phi$)

In almost all electrical machines (like generators or motors), we don't use a single loop of wire; we use a coil made up of many loops or "turns".

Magnetic Flux Linkage ($N\Phi$) is the term used to describe the total amount of magnetic flux that passes through *all* the turns in a coil.

Definition and Formula

If a coil has $N$ turns, and the flux passing through each turn is $\Phi$, the total flux linkage is:

$$N\Phi$$

Since $\Phi = BA \cos \theta$, the general formula for flux linkage is:

$$N\Phi = BAN \cos \theta$$

• \(N\) is the Number of turns in the coil (unitless).
• $N\Phi$ is measured in Webers-turns (Wb turns), though often just Webers (Wb) is used, as $N$ is a dimensionless count.

Did you know? The change in this flux linkage ($N\Phi$) over time is exactly what produces the voltage (e.m.f.) in a generator! This is Faraday's Law, the next big concept!

Application: Rotating a Rectangular Coil

A classic application is a rectangular coil of area $A$ with $N$ turns rotating uniformly in a uniform magnetic field $B$.

1. Start Position (Maximum Flux Linkage): When the coil surface is perpendicular to $B$, $\theta = 0^\circ$.
$$N\Phi_{\text{max}} = BAN$$

2. Rotated Position: As the coil rotates, the angle $\theta$ between the normal and $B$ changes over time. If the coil rotates at an angular speed $\omega$, then $\theta = \omega t$.
$$N\Phi = BAN \cos(\omega t)$$

3. Zero Flux Linkage: When the coil surface is parallel to $B$, $\theta = 90^\circ$. Flux linkage is instantaneously zero.
$$N\Phi_{\text{zero}} = 0$$

This sinusoidal relationship ($BAN \cos(\omega t)$) perfectly describes why AC generators produce a smooth, wave-like alternating current.

Key Takeaway

Flux Linkage ($N\Phi$) is the total magnetic interaction of the field with a multi-turn coil. For a rotating coil, $N\Phi$ varies sinusoidally, which is the basis for generating electricity.