d)   i) A sphere of radius a contains a uniform charge Q distributed uniformly throughout its volume. How much charge is contained in a concentric sphere of radius r (where r < a)?

ii) Using the result from Q1.d)i) and Gauss's law, show that the magnitude of the electric field E within the sphere changes with radius ras

iii)   What is the direction of the electric field E?

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2.    a)    i)       Explain what is meant by the term displacement current.

ii)     By considering Ampere's law and the magnetic field in the vicinity of a charging  capacitor,  show  that  there  is  a  need  to  introduce  the displacement current.

iii)     Show that the displacement current is essential for the derivation of the propagation of electromagnetic waves by demonstrating that, in free space, the magnetic field, B, obeys the wave equation.

b)   A magnetic field wave propagating with wavevector, k, and angular frequency, ”, maybe described by

i)       By writing B, k and r in terms of their rectangular components, or otherwise, show that

ii)      Show that the  electric field, the magnetic field and the direction of propagation are mutually perpendicular.

iii)    The Poynting vector S is defined (in free space) to be

What physical property does the Poynting vector represent?

c)    A magnetic field is given by the expression:

Determine the relationship between constants a and b.

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