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Physics for Industrial Design with Peter Eyland

Lecture 4 Magnetic Induction

In this Lecture:

• magnetic induction is defined,

• the force on a charge moving in a magnetic field is given,

• the right hand screw rule is explained,

• the equivalence of moving charge with current is derived, and

• the Hall effect is introduced.

**Magnetic Field Strength**

The magnetic field strength, **B**, is called the *magnetic induction* or the *magnetic flux density* and includes the effect of the magnetisation of the material it passes through. (Another quantity, the *magnetic intensity*, **H**, does not include magnetisation.)

To contrast things, the strength of an *electric* field () is measured by the force per unit charge that a small positive test charge will experience when placed in the electric field.

However, the *magnetic induction* () is measured by the force per unit charge *and* the speed of the charge at right angles to the field lines.

This is because the *measured* magnetic field really interacts with the *induced* magnetic field that comes from the moving charge, so the speed has to be included.

**The force on a charge moving in a magnetic field **

The magnetic force is called the *Lorentz* force and is written:

**F** = q.**v **x **B**

Where the "*x*" repesents the product of two vectors.

The *size* of the force on a charge moving in a magnetic field depends on how much of the speed is at right angles to the field lines.

When a charge is moving at an angle to the field then *v*sin*q* gives the speed at right angles to the field.

The size of the force is:

When** F** is in Newton,

The *direction* of the force is given by the right hand screw rule.

**A Sinister Plot?**

Designers make things for right-handed people e.g. corkscrews, screws, lipsticks, shower taps, and caps on bottles.

This is because the majority of people are right-handed and it is cheaper not to have two sets of machinery.

DNA is generally like a right-handed corkscrew but left-handed DNA does occur.

There is one big asymmetry in nature. In beta decay when an atomic nucleus decays through the *weak* nuclear force to form an electron and a neutrino, the neutrino is *always* like a left handed corkscrew.

The famous physicist W. Pauli wasn't impressed by this and commented that "God is not weakly left-handed". This has led to a proposal for a parallel universe of mirror galaxies to restore symmetry.

When Franklin talked about excess (+) and deficiency (-) of electric fluid for particular situations, the choice of plus and minus was arbitrary. He might have chosen the other way around and then we would have atoms with positive electrons around a negative nucleus.

Magnetic field, force and current directions are related by a right hand rule and not a left hand rule because of Franklin's choice.

**The Right Hand Screw rule**

When fixing a screw into a wall or other object, you have to rotate the screwdriver *clockwise* and the screw moves* perpendicularly away* from the plane of rotation and into the wall or object.

To get a screw out, you rotate the screwdriver *anti-clockwise* and the screw moves* perpendicularly towards* you.

The same kind of thing happens with corkscrews, lipsticks, shower taps, and caps on bottles.

In mathematical terms, rotating from the x-axis around to the y-axis gives a direction along the z-axis.

**Lorentz force direction**

To find the direction of the force on a positive charge moving in a magnetic field, you rotate from the direction of the speed towards the field direction.

The direction of the force will be at right angles to the rotation plane with clockwise rotations giving directions away and anticlockwise rotations giving directions towards you.

Example 1

A positive charge of 2μC is travelling horizontally and 20^{0} North of East at 30 m.s^{-1} when it enters a magnetic induction of 0.5 T that is pointing horizontally East.

Find the initial force on the charge.

Plan view (i.e. from above):

Fixing the tails together and rotating horizontally from speed towards field is clockwise in this case.

Hence the direction of the force is perpendicularly away from the plane of speed and field, i.e. vertically downwards.

Example 2

A positive charge of 5μC is travelling at 8 m.s^{-1} horizontally 40^{0} South of East and enters a horizontal magnetic field of 0.2 T 6^{0} East of North.

Find the initial force on the charge.

Fixing the tails together and rotating horizontally from speed towards field is anticlockwise in this case.

Hence the direction of the force is perpendicularly up from the plane of speed and field, i.e. vertically upwards.

Example 3

A positive charge of 8μC is moving horizontally North at 12 m.s^{-1} and enters a magnetic field of 0.6 T pointing vertically down.

Find the initial force on the charge.

Rotating from horizontally North to vertically down is clockwise, so the force is away towards the West.

**Circles, Spirals and Helices**

Notice that the questions ask for the *initial* direction of the force.

This is because the force is always at right angles to the plane of speed and field, so if the speed doesn't change and the charge doesn't leave the field then it will move in a circle.

The centripetal force is provided by Lorentz force, so the radius of the circle can be found.

If the speed slows then the charge will spiral inwards, but if it speeds up it then it will spiral outwards.

If the charge moves at constant speed but the speed is inclined to the magnetic field then the path will be a helix.

(Work out the possible charge signs and directions for this helical motion.)

Example

An electron has a mass of 9.1 x 10^{-31} kg is in a magnetic field of 0.25 T. The electron is moving with speed 100 km.s^{-1} at right angles to the field so that it travels in a circle.

Find the radius of the circle.

**A moving charge is a current**

Since

This says that a small quantity of moving charge has the same effect as a small length of current (called a *current element*).

Hence magnetic induction can be (more easily) measured by the force per unit length on a current element at right angles to the magnetic field lines.

The unit of magnetic induction is the Tesla and 1T = 1 N.m^{-1}.A^{-1}.

In general, the force on a current element in a magnetic field is given by:

*d F= I.dl x B*

The size of the force is given by:

*dF= I.dl sinθ B*

where *θ* is the angle between the current and the field.

The direction of the force is given by the right hand screw rule, rotating from *I.dl* to *B*.

Example

A straight wire of length 50mm has an electric current of 20A flowing in a direction N 30^{0} E.

The current is in a uniform magnetic field of induction 0.3T pointing vertically upwards.

Find the force on the wire.

The eye position is at right angles to the vertical plane of the speed and field.

Rotating in that plane from the speed to the field is anticlockwise so the force is towards the eye.

**The Hall effect**

Sometimes the current in a *semiconductor* will be a flow of positive charge, sometimes it will be negative charge, and sometimes both.

The kind of charge that flows can be found from the Hall effect.

In the Hall effect a current flows through a rectangular piece of semiconductor which is placed in a magnetic field as shown in the diagram.

Here the charges are negative so the *current* is opposite to the flow, i.e. from right to left.

The negative charges moving with their *drift speed*, *v*, are pushed to the upper side by the Lorentz force from the magnetic field. (The drift speed is faster in semiconductors than metals.)

The negative charge accumulated at the top creates a transverse electric field.

A balance will be reached when the force from the transverse electric field equals the Lorentz force.

The balance gives a potential difference called the *Hall voltage.*

If the moving charges are positive then for the *same current direction*, we have:

Here the potential difference is the opposite, so the sign of the potential difference tells which sign charges are moving.

If the current is equal positive and negative charges flowing in opposite directions the Hall voltage may be zero.

Most metals indicate that they have a flow of negative charge, but some (Zn and Cd) indicate positive flow.

Example

A block of semiconductor material that is 3mm in transverse width has a magnetic field of 0.5 T vertically down through it. The Hall voltage is measured as 90 mV and the carriers are all of one sign. Find the drift speed of the carriers.

*Summarising:*

: magnetic induction is measured by the force per unit charge and the speed of the charge at right angles to the field lines.

The magnetic force (Lorentz force) is: **F** = q.**v **x **B**

size: where *q* is the angle between * v* and

direction: right hand screw rule rotating from **v** towards **B**.

Charges moving in magnetic fields tend to move in circular arcs, spirals and helices.

The force on a current element in a magnetic field is given by:
*d F= I.dl x B*

The Hall effect is produced by a magnetic force pushing the charge carriers sideways, giving a potential difference *V _{H} = d.v_{D}.B*

The sign of the Hall potential difference indicates whether the (majority) carriers are positive or negative.

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