Two coaxial solenoids are wound one over the other on a common cardboard cylinder of negligible thickness and length 40cm. The outer solenoid has N1=1000 turns and carries current I1=3.0A; the inner solenoid has N2=500 turns and carries current I2=2.0A in the opposite direction to I1. Assume both solenoids are long compared with their diameters and that their fields are confined within the common cross-section.
a. (i) Derive an expression for the resultant magnetic field at the common central region of the two solenoids in terms of N1,N2,I1,I2, and their common length L.
b. (ii) Evaluate the magnitude and direction of the resultant magnetic field for the given numerical values.
c. (iii) Explain briefly how reversing the direction of I2 could be used to nullify the field produced by I1.