ELECTROMOTIVE FORCES

Moving charges give rise to magnetic fields.

Magnetic field in current carrying wire (using conventional current direction):

Magnetic field in flat circular coil:

Magnetic field due to solenoid:

Right-Hand Grip Rule (following conventional current flow):

In wire: thumb à current, fingers à field

In coil: thumb à field, fingers à current

Fleming’s Left Hand Rule:

Current-carrying wire in a magnetic field:

(thus if wire is parallel to field, no force on wire)

Moving charge in magnetic field:

(thus if charge is moving parallel to field or stationary, no force on charge)

Charge will move in circular motion. Velocity is always at right angle to force F, thus work done by force is always zero.

In reality the path is a spiral rather than a circle. The particle loses energy and thus its speed decreases and hence its radius decreases.

Kinetic energy of particle does not change (ideally), but momentum of the particle is changing (changing velocity/direction).

To remain undeflected, resultant force on electron must be zero. This happens when the force due to the magnetic field is equal and opposite to the force due to the electric field. Force due the electric field is upwards/down/left/right in the opposite direction (of the arrow) as the electrons are negatively charged. Force due to the magnetic field is in the opposite direction upwards/down/left/right due to Fleming’s Left Hand Rule.

1 Tesla is the flux density of a field in which a conductor of length 1m carrying a current of 1A is at right angles to a field experiencing a magnetic force of 1N. .

For square loop of wire side s falling at speed v downwards through magnetic field of flux density B (into the paper), normal to the plane of the loop:

(Induced current only in one direction to the right – using LHR, hence only through arm of loop perpendicular to velocity at which loop falls)

(where A is cross-sectional area of loop) 4s because resistance must be calculated with regards to WHOLE of the square loop, not just one arm.

Hence

And thus Force acting on loop due to electromagnetic induction:

Since induced current only flows through one arm of the loop(?), hence force due to electromagnetic induction only acts on one arm (the arm perpendicular to the velocity) in the upwards direction (using LHR if loop falling downwards and magnetic field into paper)

Induced emf is not constant even when speed of rotation of coil is constant: The amount of magnetic flux linkage with the coil changes as the coil rotates due to the changing angle the coil makes with the magnetic field.

When the magnetic flux linking a wire loop changes sinusoidally with time, the emf induced in the loop changes sinusoidally out of phase with the changing flux by a quarter period (cosine curve).

Induced Electromotive Force:

Occurs when a conductor is moved through a magnetic field (relative motion between a conductor and a magnetic field). As wire moves, electrons experience a magnetic force and a p.d. is generated along the wire. This cause an electric field E that will produce an electric force on the electrons in the opposite direction of the magnetic force.

Magnetic flux: The magnetic flux through an element of area perpendicular to the direction of a magnetic field is the product of the magnetic field strength and the area of the element.

Where is the angle between B and the surface normal to A. Hence when the coil is parallel to field, , magnetic flux is zero.

Magnetic flux linkage: The total magnetic flux through N turns of coil, ,

OR moving conductor/rod in magnetic field

 

Faraday’s Law: The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux linkage, .

Graph of magnetic flux in a rotating coil against angle between magnetic field and normal to the coil: Gradient = emf induced (rate of change of magnetic flux)

Lenz’s Law: The direction of the induced emf is such that if an induced current were to flow, it would oppose the change in magnetic flux that caused it.

Direction of Induced current: Use Fleming’s RIGHT hand rule (opposes change). External field and field due to induced current are NOT the same.

For current wire through a field where magnetic flux is increasing at a constant rate in time t, emf induced during this time t is constant, not increasing from 0 to emf value.

Current is increasing at constant rate, magnetic flux is increasing at constant rate, emf induced is a constant.

Keyword: time-changing flux

Alternating current generator:

As coil rotates in magnetic field (due to external force), flux linkage of coil changes with time. According to Faraday’s Law, emf is induced causing a current to flow, generating electrical energy. If rotation is at constant speed, induced emf is sinusoidal.

Increasing speed of rotation à Reduces time period of oscillation, and increases frequency of oscillation à Rate of change of magnetic flux linkage increased à Amplitude and frequency of induced emf increased

Root-mean-square value of an alternative current/voltage: The value of the direct current/voltage that dissipates power in a resistor at the same rate.

Transformer:

Alternating pd across the primary coil creates an alternating current within the coil and hence an alternative magnetic field in the iron core. This alternating magnetic field (changing magnetic flux linkage) links with the secondary coil and induces an emf via mutual induction. Value of the induced emf depends on the rate of change of magnetic flux linkage, which in turn depends on turns ratio of primary and secondary coils.

Why must A.C. be used?

-       For emf/current to be induced in secondary, flux must be changing in the core

-       This changing flux is caused by varying the current in the primary coil

Turns ratio:

For ideal transformers (100% efficient), !!!!!!

Mean Output power

Reasons for power losses in transmission lines: Wires do not have zero resistance, hence dissipate power, P=I2R. Over long distance, especially if large amounts of power being distributed, current used is high (P=IV), power wasted becomes significant.

Solution: Step-up transformers used to increased voltage at transmission stage, so that only small current flows hence power loss reduced. Step-down transformers then used at the end.

Disadvantage of high voltage:

-       Requires higher current, cables must have a lower resistance hence must be thicker. Such cables are more expensive and occupy more space.

-       Or, heating element must be physically larger, hence more expensive and occupies more space.

Problem: Dangerous high voltages used. Induced current in body may be linked to child cancer rates. Depend on current density, frequency and length of exposure.

Reasons for power losses in real transformers:

-       Resistive heating in wires of primary and secondary coils

-       Eddy currents circulating within core in plane normal to the flux, cause resistive heating in the core

-       Hysteresis losses in the core as a result of cycle of magnetic changes taking place

-       Leakage flux as not all of the magnetic field can be concentrated within the core; leakage flux intercepts nearby conductive materials such as transformer’s support structure, giving rise to eddy currents and heat loss.

Solution: To prevent current from flowing in the core, laminated core used, made up of individually electrically insulated thin strips

Why must laminated core be used? (IB question)

-       Changing magnetic flux induces eddy currents, this ensures induced currents in the core are kept small (do not allow reduces/prevented)

-       To reduce heating and energy losses (do not allow mere “eddy current losses”)

Extra low-frequency alternating magnetic fields: Originate from electrical appliances and electric power lines, can induce a changing electric field that will in turn induce small alternating currents within any conductor such as human bodies.