7.5.4

# Magnetic Flux & Flux Linkage

Test yourself

## Magnetic Flux Definitions

Three terms that are closely related but different are magnetic flux, magnetic flux density and magnetic flux linkage.

### Definitions

• Magnetic flux is a measure of the number of field lines passing through a region of space.
• Magnetic flux density is a measure of the number of field lines passing through a region of space per unit cross-sectional area.
• Magnetic flux linkage is the product of the magnetic flux and the number of turns on a coil through which the field passes.

### Different flux density

• The two loops in the diagram have the same flux but different flux densities since area Y is larger than area X.

### Different flux

• The two loops in the diagram have the same flux density but different flux since area Y is larger than area X.

### Different flux linkage

• The two loops in the diagram have the same flux and flux density, but as loop Y has twice the number of turns as loop X, the flux linkage in Y is twice that of loop X.

## Magnetic Flux and Flux Linkage

The formulae for magnetic flux and magnetic flux linkage are:

### Magnetic flux, Φ

• The magnetic flux is the amount of magnetic field passing through a surface.
• The equation for magnetic flux is:
• $\Phi = BA \cos \theta = B_\perp A$
• Where $\phi$ is the flux, B is the field strength, A is the area and theta is the angle between the normal of the surface and the field.

### Magnetic flux linkage, nΦ

• The magnetic flux linkage is the amount of field passing through a coil of wire. It is the flux 'linked' to a wire.
• The equation for the flux linkage is:
• $N \Phi = BAN \cos \theta$
• Where N is the number of coils in the wire.

## Magnetic Flux Linkage Experiment

This experiment investigates how magnetic flux linkage varies with the angle between a search coil and the magnetic field direction. This is done using a search coil and an oscilloscope.

### Apparatus

• Set up the apparatus as shown in the diagram.
• The search coil will experience a flux linkage as determined by the formula:
• $N \Phi = BAN \cos \theta$.

### Underlying theory

• The magnetic field inside the coil will be uniform and running parallel to the length of the spring (across the page horizontally in the diagram).
• As the current is a.c., there will be a varying magnetic flux density through the search coil, and hence a varying flux linkage.

### Underlying theory 2

• This will induce an EMF in the search coil which is detected by the oscilloscope.
• The time base (time axis) of the oscilloscope is turned off, so that the a.c. induced gives a vertical line on the screen. The amplitude of this line (from the peak to the middle) gives the magnitude of the induced EMF.

### Method

• The search coil can now be rotated so that the angle between the plane of the coil and the magnetic field lines changes.
• By recording the amplitude of the induced EMF every 10 degrees, a graph of EMF induced against angle can be plotted, showing the relationship between the flux linkage and the angle between the normal to the plane of the coil and the magnetic field lines.