(a) Given that the magnetic field generated at the centre of an air-cored solenoid is given by B = µonI, where n is the number of turns per unit length, and I is the current, show that the self-inductance of a long solenoid of length I and area A is equal to µon²lA.
(b) A solenoid with I = 10 cm and diameter 1 cm is connected to a battery via a switch. The solenoid has 2000 turns per metre and is made of copper wire with resistivity 1.7 × 10-8 ohms m.Estimate the time taken after the switch is closed for the magnetic field to reach 90% of its maximum value. (Assume that the wires in the solenoid are packed as close together as they can be.)
(c) How would your answer to (b) change if the core of the solenoid were filled with iron with relative permeability of 5000?
(d) Consider the case where the battery has an EMF of 12 V and there is a resistor with R = 1000 2in series with the solenoid. Find the energy stored in the solenoid after the current has settled to a constant value for: (i) the air-cored solenoid, and (ii) the iron-cored solenoid.
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