A thin plastic rod of length L has a positive charge Q uniformly distributed along its length.*ALd**B (a) Taking V = 0 at infinity, integrate the potential due to individual sources charges on the rod (i.e.use V = f k dq/r) to find an expression for the electrostatic potential at point A, a distance dabove the midpoint of the rod. (Treat the rod as a line of charge.) Use integral tables, WolframAlpha, your calculator, or whatever you need, but you do need to actually get the result of theintegration.(b) Evaluate the result for Q = 1 x 10-9 C, L = 0.080 m, and for d = 0.010 m.(c) Compare to 10 point charges in a similar geometry, distributed along the length L, with the sametotal charge as before, Qtotal = 1 x 10-⁹ C, so each point charge has q = Qtotal/10 = 1 x 10-¹⁰C. Use the same values for L = 0.080 m, and for d = 0.010 m.(d) Check the analytic result for the continuous source by examining the potential far away, atdistances d much greater than the length L of the rod. When d > L, we expect that we can nolonger discern the particular size and shape of the charge distribution and that instead it willresemble a point charge. Thus, the expression for the electric potential should also resemblethe expression for the electric potential of a point charge. Check this prediction at a few testpoints by evaluating your expression from part (a) using values of d» L. For example, youmight calculate the exact and approximate results using d=1.0 m and again using d =3 m, orany distances that are much larger than the length. Use at least two different test points so youcan see that the result has the right dependence on the distance - if you triple the distance, asI did in my suggested distances, what should happen to the potential?FOR E WORK ONLY: Show that your numerical tests are generally true of your results - youwill need to use the fact that for x < 1, ln(√1 + x² + x) ≈ ln(1 + x) = x. Show that this isnumerically true for x=0.01, and even better for x=0.001.Briefly comment on your results. (Note that we do not want to consider the purely mathematicallimit d→ ∞ here, but a situation in which d is large compared to the length L of the rod.)Should the exact answer (your answer to part a) be larger than the approximation we just made,smaller than it, or does that vary by distance? Present a physics-language argument, and thencheck mathematically (numerically if you wish, algebraically or graphically if you can.)(e) Should the potential at A be larger, smaller or equal to the potential at a test point B, adistance d to the right of the right end of the rod? Justify using a physics argument, then findan expression for the potential at B and check that the math answer agrees with your physicsone. You may check numerically (using the same values for L, d, and Q as before) or analytically(comparing the general expressions in terms of variables.)

Solution: a). Let us consider a line charge of a given length and charge as shown below. Also,…

Answered: A thin plastic rod of length L has a…

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A thin plastic rod of length L has a positive charge Q uniformly distributed along its length. *A