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Physics for Industrial Design with Peter Eyland

Lecture 5 Investigating magnetic fields

 

In this lecture:

• the Hall probe is introduced,

• the Biot Savart definition is given and its use demonstrated,

• Ampere's rule is given and its use demonstrated,

• the magnetic induction in "short wide" and "long narrow" solenoids is calculated,

• the force/length between two long straight current carrying wires is calculated.

 

Hall probe

Magnetic induction can be experimentally measured with a Hall probe.

A Hall probe has a small piece of semiconductor with a constant current along it.

The transverse potential difference measures the magnetic induction.

The Hall potential at a point will vary with the orientation of the probe so the direction of the field will be given from its maximum value and polarity.

This part of the course looks at the experimentally measured shape of magnetic fields in a few simple cases and how the magnetic induction can be calculated for them.

 

The Magnetic Induction from the current in a long straight wire

From Hall probe measurements, the current in a long straight conductor induces a magnetic field that is

• constant around cylindrical areas centred on the current and

• the size of the induction falls off inversely with the radial distance from the centre.

The "long straight current" is a sum of current elements laid out end to end in a line.

The cylindrical magnetic field is the sum of the fields from these aligned current elements.

However symmetry considerations mean that a tiny current element will produce a spherical magnetic field around it.

The cylindrical field is then the result of spherical fields squashed together: like line of squashed balloons makes a cylinder.

 

 

Biot - Savart's definition

Jean Baptiste Biot and Félix Savart wrote down the magnetic field that would be induced by a current element.

The field at a point in space must be

• proportional to the size of the current element,

• proportional to how far you are from it, and

• spread over the surface of a sphere.

The field lines are in the spherical surface, i.e. at right angles to the plane of the current and the radial direction to the point.

This gives:

B is the magnetic induction in Tesla and its direction is given by the right hand screw rule from idl to r.

 is the proportionality constant for space, called the magnetic permeability in T.m.A-1 (Tesla.metre/Ampere).

For air/vacuum it is equal to 4π x 107 T.m.A-1

 

i is the current in Ampère

is the length of the current element in m

r is the distance in metre from the current element to the point in space where B is calculated.

This definition (theoretically) enables us to calculate the magnetic field from any shape of current carrying wire.

 

The Magnetic Induction at a radial distance from the current in a long straight wire

This is the kind of thing that you would do to calculate the magnetic induction.

 

The result has typical cylindrical symmetry:

• it depends on the originating current, and

• it is distributed around the circumference of a circle centred on the wire, and so falls of inversely with radial distance.

 

Example

A magnetic induction of 20 μ T is measured at a point 120 mm at right angles away from a long straight current carrying wire.

Find the current in the wire.

 

The Magnetic Induction at the centre of a circular loop of current.

From Hall probe measurements, a current flowing in a circular loop produces what is called a dipole field.

The diagram gives a cross sectional view.

It looks a lot like the Earth's magnetic field because the Earth's main field is about 90% dipole.

 

To get an idea of what happens with the magnetic induction at the centre of the loop, it is here divided into 20 circular arcs.

Take the arc indicated and put the tails of the arrows together. They will be at 900 as shown.

Rotating from the current towards the radius is anticlockwise, so the induction is towards you, i.e. vertically up.

Each of the 20 arcs will give the same direction and size because they are just rotated at 180 to each other.

Having seen the contributions from arcs of current, the Biot Savart definition is now integrated for infinitesimal lengths of loop:

The magnetic induction at the centre of a current carrying loop of conductor depends

• on the size of the originating current, and

• the width of the loop.

 

The Magnetic Induction at the centre of a short wide solenoid.

A few closely wound coils gives a short wide solenoid, i.e. the length is shorter than the diameter.

 

This will give a dipole like field, as shown by Hall probe measurements, so the magnetic induction at the centre can be taken to be the induction for one loop multiplied by the number of loops.

 

For N loops:

 

Example: (short wide coil)

A solenoid with 150 turns of insulated wire has a length of 20 mm and a diameter of 150 mm.

It carries a current of 6 A.

Find the magnetic induction at the middle of the solenoid.

 

 

Ampere's rule

In both of the examples above, the magnetic induction times a length is proportional to the originating current.

Things like this led Andre Marie Ampere to propose the following rule.

For a closed loop in space, the scalar product of magnetic induction and distance around a closed loop is proportional to the enclosed current(s):

 

The Magnetic Induction at a radial distance from the current in a long straight wire

Reworking the first example with Ampere's rule:

Take a circle of radius r centred on the wire.

The magnetic induction times the length around the circle is proportional to the current through the wire.

The result is the same as before, but the calculation is much simpler.

 

Magnetic induction at the centre of a long narrow solenoid

A long narrow solenoid has a length much greater than its diameter.

From Hall probe measurements, the magnetic field is mostly constant (direction and size) inside the solenoid and falls off quickly at the ends.

The field outside is close to zero and it is presumed that the magnetic field lines form closed loops at very large distances from the solenoid.

The solenoid shown below has a length L with N total loops (or turns).

Take the square of length x and use Ampere's rule around it.

Of the four sides, only one is lined up with the field and contributes.

Two sides are at right angles and so contribute nothing.

One is outside where the field is effectively zero and so contributes nothing.

Where n is the number of turns per length.

Because it is narrow, the diameter is not important and the magnetic induction is:

• uniform inside the solenoid,

• proportional to the originating current, and

• proportional to the number of turns per unit length.

 

Example: (long narrow coil)

A coil of 1000 turns of insulated wire is 75 mm in length and has a diameter of 40 mm.

It carries a current of 12 A.

Find the magnetic induction through the centre of the coil.

 

 

The force between two long parallel current carrying wires

This can be done in a couple of ways,

• by considering the interaction of the two fields produced by the currents, or

• by considering one as the producer of a magnetic field and the other as a current in that magnetic field.

The opposing arrows do not merge in the middle so one jumps to the other side and the currents are pulled inwards.

The parallel arrows do not merge in the middle so the currents are pushed apart.

 

Now take the current on the left as the provider of the magnetic field and the current on the right as experiencing a force because of that magnetic field.

Looking sideways from the right hand side, the rotation from iR to B is clockwise from the horizontal up to the vertical, so the force is "away", i.e. the force is towards the other current.

 

Looking sideways from the right hand side, the rotation from iR to B is anticlockwise from the horizontal down to the vertical, so the force is "towards", i.e. the force is away from the other current.

 

Example

Two long parallel wires are 40mm apart and have currents in the same direction.

If the currents are 20A and 30A, find the force per unit length that attracts them.

 

 

 

Summarising:

The Hall probe measures both the size and direction of magnetic induction.

The Biot Savart definition is:

Ampere's rule is:

Radially from a straight current:

At the centre of a circular current:

In a "short wide" solenoid:

In a "long narrow" solenoid:

 

The force/length between two long straight current carrying wires is:

 

 


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