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**Magnetic Fields 1**

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

__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.

__Flux Density B __

We can refine the idea of flux by making the area unity (1m^{2}). This introduces a new concept - **magnetic flux density**** B** .

For *normal* area (area at right angles),

**total magnetic flux = flux density x area **

__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**.

So the total normal flux over an area * A* at an angle

**θ**to the field is given by :

__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.

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**.

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

__Flux Density for a Straight Wire__

The diagram below illustrates the flux density * B* at a point

**P**a distance

*away from the wire.*

**a**

The magnetic flux density * B *is described by the equation :

where * μ_{o}* is the permeability of free space.

__Flux Density for an Infinitely Long Solenoid__

The diagram illustrates the flux density * B *in a solenoid with

*turns and coil current*

**n***.*

**I**

The magnetic flux density * B *is described by the equation :

where * μ_{o}* is the permeability of free space and

*is the number of turns per unit length of the solenoid.*

**n**

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**

*, with units of*

**H****amp-turns/metre**(

**Am**) .

^{-2}

**NB** turns * n* has no units

__Uniform Magnetic Fields - Helmholtz Coils__

A single plane coil of radius * r *, turns

*and current*

**N***produces magnetic flux density*

**I***at its*

**B****centre**.

where * μ_{o}* 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

*is the coil radius.*

**r**

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

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

where * μ_{o}* 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* ,

*and*

**β***particles from each other.*

**γ**

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