Question

Now the quest on you've all been wait ng for - determining the f eld due to an extended, cont nuous object using integrals. Consider a thin, charged rod of length, L It lies along the x-axis with one end at x =0 and the other end at x-L. It has a non-uniform linear charge density given by the funct on (x)-0+Ax where N0 and A are both posit ve constants. (a) What are the units of X0 and A? Explain. (b) Suppose that N0 and A are related such that A 20/L. What is the charge density of the rod at x-L? Is the lef (x 0) end of the rod posit vely or negat vely charged? Is the right (x-L) end posit ve or negat ve? (c) What is the total charge on the rod? You shouldn't need to do any calculat on. If you've understood (x) then the answer should be "obvious". (d) Consider any point, P, lying somewhere on the x-axis to the right of the rod. It might be helpful to def ne x at point P asxL+ . Which way must the electric feld at P be point ng? In part cular, do you need to worry about calculat ng both components of the electric feld, and is it already clear what the signs of the components must be? (e) Draw a diagram showing the rod, the axes, and an arbitrary "bit" of the rod. Use this diagram to def ne the symbols you will need to solve this problem. In part cular, you need a variable which specif es the locat on of each "bit" of the rod which you will eventually integrate over, and an expression for the amount of charge on the "bit". Don't cut corners on this diagram! A carefully drawn diagram will help you to understand how to solve the problem and will help the reader (...marker...) to understand ur solut on vo (f) Write the approximate E-f eld due to the one "bit" of rod (g) Use your expression for the E-f eld due to the "bit" to build the integral which will give you the E-f eld at point P. In part cular, be careful of the fact that the two halves of the rod have dif erent signs of charge (h) Evaluate your integral any way you want (by hand, using a computer applicat on, using a fancy T calculator, whatever...). You should check the units of your fnal expression to be sure that you have actually calculated an electric feld (i) BONUS: Consider a point Q, anywhere on the posit ve y-axis. Find the electric f eld (vector...) at point Q.

Answer 1