Magnetism and Induction

What to know:

• Field lines run north to south outside a magnet and south to north inside it; they form closed loops.
• Two field lines never cross: the field at a point has only one direction. Lines closer together mean a stronger field. They are not always parallel.
• Diamagnetic materials are weakly pushed away; paramagnetic ones are weakly pulled in; ferromagnetic ones are strongly pulled in and can be made into magnets.
• Ferromagnetism needs tiny regions (domains) of lined-up atoms held in a solid, so liquids and gases are never ferromagnetic. Above its Curie temperature a ferromagnet becomes paramagnetic.
• The Earth's field at a place is fixed by three things: declination (angle between geographic north and magnetic north), dip or inclination (angle the field makes with the horizontal) and the horizontal component. Declination and dip together give its direction.

What to know:

• Straight wire: field lines are concentric circles round the wire; the field is weaker farther away.
• Circular loop: near the centre the lines are almost straight.
• Long solenoid: inside, the field is uniform and the lines are parallel straight lines; outside, it looks like a bar magnet's field. A soft iron core makes it an electromagnet.
• The field grows with the current and the number of turns.
• Two parallel wires: currents the same way attract, opposite ways repel. Close wires with equal and opposite currents: midway between them the two fields add; far away they almost cancel.
• Faraday's law gives only the size of an induced emf; Lenz's law gives its direction.

The rules:

• On a charge: $F = qvB\sin\theta$, with $\theta$ the angle between velocity and field. Maximum at $\theta = 90^{\circ}$; zero when parallel or anti-parallel.
• On a wire: $F = BIl\sin\theta$.
• Direction: Fleming's left-hand rule, or $\vec{F} = q\,\vec{v} \times \vec{B}$. For a positive charge the "current" points along its velocity; for an electron, turn the answer round.
• The force is at right angles to the velocity, so it does no work: the speed stays the same and the path curves (a circle when $\theta = 90^{\circ}$).
• The electric motor works on the force on a current-carrying coil in a field.

The rules:

• Faraday's law: induced emf $= N \times$ rate of change of magnetic flux. Faster change, bigger emf.
• Lenz's law: the induced current opposes the change that causes it (hence the minus sign).
• Fleming's right-hand rule gives the direction of the induced current. A generator turns mechanical energy into electrical energy.
• Transformer (AC only): $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p}$. Step-up: more turns on the secondary. In an ideal transformer $V_pI_p = V_sI_s$, so stepping the voltage up steps the current down.
• A transformer heats up from its losses: heating of the windings ($I^{2}R$, copper loss) and losses in the core (hysteresis and eddy currents). The core is made of thin laminated sheets to cut eddy currents; big ones are cooled by oil.