Question: consider the charged rod and recall that it lies along...
Consider the charged rod and recall that 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 function
λ(x) = −λ0 + Ax ,
where λ0 and A are both positive constants.
(a)consider any point, P, lying somewhere on the x-axis to the right of the rod. It might be helpful to define x at point P as x = L + l. Is the potential at point P positive or negative? Explain how you know.
(b) Should the potential decrease or increase as we look farther to the right from point P? Explain.
(c) Draw a diagram showing the rod, the axes, and an arbitrary “bit” of the rod. Use this diagram to define the symbols you will need to solve this problem. In particular, you need a variable which specifies the location 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 to understand your solution.
(d) Write the approximate potential at point P due to the one “bit” of rod.
(e) Use your expression for the potential due to the “bit” to build the integral which will give you the potential at point P due to the whole rod. In particular, be careful of the fact that the two halves of the rod have different signs of charge.
(f) Evaluate your integral any way you want,You should check the units of your final expression to be sure that you have actually calculated an electric potential.
Please answer (f) and (e)