(a) Magnetic field lines show the directions (at every point) which a small magnetised needle takes up (at that point). Do the magnetic field lines also represent the lines of force of a moving charged particle at every point?
(b) Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid. Why?
(c) If magnetic monopoles existed, how would Gauss's law of magnetism be modified?
(d) Does a bar magnet exert a torque on itself due to its own field? Does one element of a current carrying wire exert a force on another element of the same wire?
(e) Magnetic field arises due to charges in motion. Can a system have magnetic moment even though its net charge is zero?
(f) Magnetic force is always normal to the velocity of a charge and therefore does no work. An iron nail held near a magnet, when released, increases its kinetic energy as it moves to cling to the magnet. What agency is responsible for this increase in kinetic energy if not the magnetic field?

Explanation

(a) We know, force on a charge q moving with a vel, `vecv` in a uniform magnetic field of strength `vecB` is `vecF=q(vecvxxvecB)`
`:.` Magnetic force is always normal to `vecB`
Magnetic field lines of B cannot represent the lines of force of moving charged particle.
(b) Magnetic field lines can be entirely confined to the core of a toroid because toroid has no ends. It can confine the field within its core. A straight solenoid has two ends. If the entire magnetic flux were confined between these ends, the magnetic field lines will no longer be continuous.
(c) According to Gauss's law in magnetism, magnetic flux over any surface (closed or open) is always zero, i.e., `oint vecB.vecds=0`
If monopoles existed, the magnetic flux would no longer be zero, but equal to `mu_0` times the pole strength enclosed by the surface, i.e..,
`oint vecB.vecds=mu_0m`
(d) No, there is no force or torque on an element due to the field produced by that element itself.
But there is a force (or torque) on an element of the same wire. However, for the special case of straight wire, this force is zero.
(e) Yes, a system can have magnetic moment even if its net charge is zero. For example, every atom of para and ferromagnetic materials has a magnetic moment, though every atom is electrically neutral. Again, a neutron has no charge, but it does have some magnetic moment.
(f) An iron nail is made up of a large number of atoms, in which so many electronic charges are in motion. All these charges in motion experience a magnetic force when held near a magnet. The magnetic forces do not change speed of the charges, but they do change their velocity. The velocity of centre of mass may increase at the expense of nail's internal energy. Thus internal energy of the nail is responsible for increase in kinetic energy of the nail, as a whole.