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Fields & Effects

 

Magnetic Fields 1

 

flux φ

flux density B

magnetic field straight wire

B straight wire

B solenoid

Helmholtz coils

 

 

 

fast revision: Magnetic field lines follow the direction of a free moving North Pole.

 

 

direction of a magnetic line of force

 

 

Magnetic Flux φ (phi)

 

definition: magnetic flux is a measure of the strength of a magnetic field over a given area perpendicular to it

 

The diagram below shows how the magnetic flux φ over an area A varies around the pole of a magnet.

 

 

magnetic flux around a magnet

 

 

 

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Flux Density B

 

We can refine the idea of flux by making the area unity (1m2). This introduces a new concept - magnetic flux density B .

 

 

flux related to flux density

 

 

For normal area (area at right angles),

 

total magnetic flux = flux density x area

 

 

flux related to  flux density - equation #1

 

Units

 

The unit of flux is the Weber (Wb) and the unit of flux density is the Tesla (T).

 

A flux density of 1 Tesla is 1 Weber per square metre.

 

1 T = 1 Wbm-2

 

For an area A at an angle θ to the magnetic field, normal flux density has magnitude Bcosθ .

 

 

magnetic flux density and area

 

 

So the total normal flux over an area A at an angle θ to the field is given by :

 

magnetic flux

 

 

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Magnetic Fields around Current-Carrying Conductors

 

The magnetic field around a current-carrying wire is a series of concentric field lines.

 

The field is not uniform.

 

The lines are not evenly spaced, being tightly packed close to the wire and widely spaced away from it.

 

 

magnetic field around a straight wire

 

 

The direction of the lines of force is clockwise in the direction of the current direction.

 

 

The field around a plane circular coil resembles the field around a short bar magnet.

 

 

magnetic field around a single coil

 

 

The field around a solenoid resembles the field around a long bar magnet.

 

 

magnetic field around a solenoid

 

 

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Flux Density for a Straight Wire

 

The diagram below illustrates the flux density B at a point P a distance a away from the wire.

 

 

B for a straight wire

 

 

The magnetic flux density B is described by the equation :

 

long wire flux density equation

 

where μo is the permeability of free space.

 

 

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Flux Density for an Infinitely Long Solenoid

 

The diagram illustrates the flux density B in a solenoid with n turns and coil current I.

 

 

B for an infintely long solenoid

 

 

The magnetic flux density B is described by the equation :

 

flux density for a solenoid

 

where μo is the permeability of free space and n is the number of turns per unit length of the solenoid.

 

 

The value of B approximates to that of a real solenoid provided the solenoid's length is at least x10 its diameter.

 

The quantity nI is of some significance.

 

nI is equal to the magnetic field strength H , with units of amp-turns/metre (Am-2) .


NB turns n has no units

 

field strength H in terms of turns and current - equation #1

 

 

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Uniform Magnetic Fields - Helmholtz Coils

 

A single plane coil of radius r , turns N and current I produces magnetic flux density B at its centre.

 

 

B for a plane circular coil

 

 

flux density for a plane circular coil

 

where μo is the permeability of free space.

 

 

Helmholtz Coils produce a region of uniform magnetic field within a discrete volume.

 

Two identical plane coils are aligned along a common axis and positioned a distance r apart, where r is the coil radius.

 

 

Helmholtz coils - diagram #1      

 

 

The current I passing through each coil is the same and in the same direction.

 

 

 helmholtz coils - diagram #2 

 

 

The magnetic flux density B in the volume of uniform field (shaded green) is given by :

 

Helmholtz coil equation

 

where μo is the permeability of free space.

 

 

Helmholtz coils are particularly useful for deflecting electron/ion beams.

 

All charged particles follow a circular path when injected into a magnetic field at right angles to their motion.

 

By measuring the radius of a path and whether the path is clockwise or anticlockwise, important information can be gleaned on the charge of a particle and its mass.

 

This method is particularly important in distinguishing α , β and γ particles from each other.

 

 

 

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