A long solenoid with 1000 turns/m has a core material with relative permeability 500 and volume 10³ cm³. If the core material is replaced by another material having relative permeability of 750 with same volume maintaining same current of 0.75 A in the solenoid, the fractional change in the magnetic moment of the core would be approximately $$\left( {{x \over {499}}} \right)$$. Find the value of x.

A uniform conducting wire of length is 24a, and resistance R is wound up as a current carrying coil in the shape of an equilateral triangle of side 'a' and then in the form of a square of side 'a'. The coil is connected to a voltage source V₀. The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is 1 : $$\sqrt y $$ where y is ................. .

A coil in the shape of an equilateral triangle of side 10 cm lies in a vertical plane between the pole pieces of permanent magnet producing a horizontal magnetic field 20 mT. The torque acting on the coil when a current of 0.2 A is passed through it and its plane becomes parallel to the magnetic field will be $$\sqrt x $$ $$\times$$ 10^$$-$$5 Nm. The value of x is .................

Two short magnetic dipoles m₁ and m₂ each having magnetic moment of 1 Am² are placed at point O and P respectively. The distance between OP is 1 meter. The torque experienced by the magnetic dipole m₂ due to the presence of m₁ is ........... $$\times$$ 10^$$-$$7 Nm.