Lecture 7 Electromagnetism

Heinrich Lenz stated this as:

The current that is induced in a closed conducting loop, by a change in magnetic flux, flows in a direction that opposes the change in flux.

This means that if the magnetic flux decreases though a loop then current will flow to try to bring the flux back up to its original value.

If the magnetic flux through a loop increases then current will flow to try to bring the flux down to its original value.

Example

A single coil of conductor with a radius 50 mm has a magnetic field of 0.3 T through it at right angles to the area of the loop.

The magnetic field is reduced to 0.1 T in 0.7 ms.

Find the e.m.f. induced in the coil.

A transitory current flows clockwise.

Example

A rectangular coil with dimensions 60 mm x 70 mm has 80 turns.

Initially it has a magnetic induction through it of 0.4 T at right angles to the area of the coil.

The magnetic induction is reduced at a constant rate for a time of 0.5 ms.

The magnetic induction after this time is 0.1 T in the same direction.

Find the e.m.f. induced in the coil during this time.

Note: If B₂ was in the opposite direction, then

Motional e.m.f.s

E.m.f.s can be induced by the motion of the loop, as shown below.

The current will flow clockwise to increase the flux inside the loop.

Alternating Currents

Notice that an e.m.f. is only generated while the flux is changing.

Rotating a loop changes the cross-sectional area that the magnetic field "sees", thus changing the magnetic flux.

Rotating a loop clockwise from above.

Example

A coil of diameter 60 mm with 150 turns rotates at 500 revolutions per minute in a uniform magnetic field of 0.7 T.

Find the peak e.m.f.

derivation:

This is a current that

• increases to a maximum in one direction, then

• decreases to zero in that direction, then

• increases to a maximum in the reverse direction, then

• decreases to zero in that direction, etc etc.

The current alternates sinusoidally in direction with frequency Hertz.

Using this result:

Eddy currents

When the magnetic induction through a large area of conductor changes, then eddy currents are produced which dissipate energy into the conductor.

This can be used for magnetic braking or heating (induction furnace).

If energy deposition is undesirable then the material can be laminated.

Transformers

Transformers (which are two coils wound around each other and an Iron core) can raise or lower the potential difference in a circuit.

Transformers can used to minimise energy losses in transporting electricity over long distances and matching load resistance so that there is maximum power transfer.

The coils are drawn separately below, for clarity, but their areas match and overlap so that the magnetic flux is the same in each turn of both coils.

The subscripts "p" and "s" stand for primary and secondary.

The primary coil is connected to the source and the secondary coil is connected to the load.

From Faraday's law:

The potential can be multipied or divided by adjusting the "turns ratio" N_s/N_p.

Note that the currents flow in opposition from Lenz's law.

In an ideal transformer there is no energy loss, and so we can write:

Example

A large city uses 100MW of power at 240V.

It is supplied from a power station 100km away through a cable which has total effective resistance of 0.01 W>.km^-1.

Find

(a) the power loss in the cable if power is transmitted from the power station at 240V.

(b) the turns ratio to step down to 240V from 330kV

(c) the current in the cable if the power station supplies power at 330kV

(d) the power loss in the cable if the power is transmitted at 330kV

Power matching transformers

Maximum power is dissipated in a load resistance when the load matches the internal resistance of the source.

The effective load seen through the transformer is given by:

Hence the effective resistance can be made equal to the internal resistance of the source by adding a transformer with the appropriate turns ratio.

Example

A TV is designed to have 75 W in the two-core flat wire to its antenna.

When used with "coax" cable the resistance in the aerial system is 300 W.

Find the turns ratio of a transformer which would supply maximum power to the TV when coax cable is used.

Summarising:

Magnetic Flux in T.m²:

Faraday's and Lenz's laws:

Alternating Currents are produced by rotating a conducting coil in a magnetic field.

Eddy currents deposit energy in large conductors when the magnetic flux through them changes.

Transformers:

Energy losses are minimised by transporting electricity at high potential then using transformers to reduce the potential and increase current at the load.

Transformers can be used to match load resistance so that there is maximum power transfer.

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