5. [15 marks] [15-25 minutes] (a) Electric charge is conserved, i.e. it can neither be created from nothing or destroyed to nothing. The amount of charge owing out of a closed surface per unit time MUST equal the decrease of the charge per unit time in this volume (and the reverse if charge is owing in, the rate of ow in must equal the rate of increase in charge inside the closed volume). Consider an arbitrary closed volume surrounded by surface S, then it is a fact of nature that because of the conservation of charge we can write: where Q is the total charge enclosed inside the volume (in C) and J is the current density (Afmz). Now this charge is the integral of the 3D charge density p of the charge distribution (Cfm3) within the volume, right, so we can write: 9= m p m p d. rafter"! 11)th e3.d32_%=_%[jpdv]. 5' twitter! Now convert the surface integral into a volume integral using Gauss's Divergence Theorem to convert the surface integral to a volume integral to show that: You have just derived a very important relation that is much more general than in electromagnetism. What have you just derived [does it have a name in the literature \". . "equation\" !? and What is its meaning? [The derivation is just 1 step (literally 1 line) + a small text explanation of why the step is taken follow the method we did when we converted the integrated form of Maxwell's Equations to their differential form. Don't forget to give the reference if you use any]. (b) Now suppose you have a steady-state situation (i.e. the charge density distribution does not change in time). What is the value of the surface integral in this case and what is the divergence of I in this case? Comment briey