Question

a.     Work out the product of the following complex numbers; note that a is a constant:

                         I.          (50+60i)(-5+12i)

                       ii.           (3ai)(a-10i)

b.     When an alternating current is sent through a wire coil, the current flow is impeded by it in the same way that a resistor resists or impedes the flow of electrical current through it. It is said to have an impedance – just as a resistor has a resistance. However, because the electrical signal also experiences a phase change, it is usual to refer to it as a complex number. In this case, we can write the impedance as

Z=2iπfL,

where f is the frequency of the current passed through the coil, L is the inductance of the coil (in units of henry, H) and i is the complex number such that i²=-1.

If the current passing through the coil has a frequency of 50Hz (i.e. f=50) and a magnitude of 12A, then we can write it as 

I=12cos(2x50πt)+12isin(2x50πt),

where t is time.

Assume that Ohm’s Law applies to coils (or inductors), in other words, the voltage v across the coil is given by V=IZ.

Calculate the impedance of this coil and the voltage across it as a complex number when L=5x10^-3H and t=0.001s. Give your answers to one decimal place.