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(a) Taking V = 0 at infinity, integrate the potential due to individual sources charges on the rod (i.e. use V = k dq/r) to find an expression for the electrostatic potential at point A, a distance d above the midpoint of the rod. (Treat the rod as a line of charge.) Use integral tables, Wolfram Alpha, your calculator, or whatever you need, but you do need to actually get the result of the integration. (b) EvaluatetheresultforQ=1x10-9 C,L=0.080m,andford=0.010m. (c) Compare to 10 point charges in a similar geometry, distributed along the length L, with the same total charge as before, Qtotal = 1 x 10-9 C, so each point charge has q = Qtotal/10 = 1 × 10-10 C. Use the same values for L = 0.080 m, and for d = 0.010 m.

(a) Taking V = 0 at infinity, integrate the potential due to individual sources charges on the rod (i.e. use V = k dq/r) to find an expression for the electrostatic potential at point A, a distance d above the midpoint of the rod. (Treat the rod as a line of charge.) Use integral tables, Wolfram Alpha, your calculator, or whatever you need, but you do need to actually get the result of the integration. (b) EvaluatetheresultforQ=1x10-9 C,L=0.080m,andford=0.010m. (c) Compare to 10 point charges in a similar geometry, distributed along the length L, with the same total charge as before, Qtotal = 1 x 10-9 C, so each point charge has q = Qtotal/10 = 1 × 10-10 C. Use the same values for L = 0.080 m, and for d = 0.010 m.

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(a) Taking V = 0 at infinity, integrate the potential due to individual sources charges on the rod (i.e. use V = k dq/r) to find an expression for the electrostatic potential at point A, a distance d above the midpoint of the rod. (Treat the rod as a line of charge.) Use integral tables, Wolfram Alpha, your calculator, or whatever you need, but you do need to actually get the result of the integration. (b) EvaluatetheresultforQ=1×10-9 C,L=0.080m, andford=0.010m. (c) Compare to 10 point charges in a similar geometry, distributed along the length L, with the same total charge as before, Qtotal = 1 x 10-9 C, so each point charge has q = Qtotal/10 = 1 x 10-10 C. 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, at distances d much greater than the length L of the rod. When d » L, we expect that we can no longer discern the particular size and shape of the charge distribution and that instead it will resemble a point charge. Thus, the expression for the electric potential should also resemble the expression for the electric potential of a point charge. Check this prediction at a few test points by evaluating your expression from part (a) using values of d » L. For example, you might calculate the exact and approximate results using d=1.0 m and again using d =3 m, or any distances that are much larger than the length. Use at least two different test points so you can see that the result has the right dependence on the distance - if you triple the distance, as I did in my suggested distances, what should happen to the potential? (e) Should the potential at A be larger, smaller or equal to the potential at a test point B, a distance d to the right of the right end of the rod? Justify using a physics argument, then find an expression for the potential at B and check that the math answer agrees with your physics one. 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.)
Transcribed Image Text:(a) Taking V = 0 at infinity, integrate the potential due to individual sources charges on the rod (i.e. use V = k dq/r) to find an expression for the electrostatic potential at point A, a distance d above the midpoint of the rod. (Treat the rod as a line of charge.) Use integral tables, Wolfram Alpha, your calculator, or whatever you need, but you do need to actually get the result of the integration. (b) EvaluatetheresultforQ=1×10-9 C,L=0.080m, andford=0.010m. (c) Compare to 10 point charges in a similar geometry, distributed along the length L, with the same total charge as before, Qtotal = 1 x 10-9 C, so each point charge has q = Qtotal/10 = 1 x 10-10 C. 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, at distances d much greater than the length L of the rod. When d » L, we expect that we can no longer discern the particular size and shape of the charge distribution and that instead it will resemble a point charge. Thus, the expression for the electric potential should also resemble the expression for the electric potential of a point charge. Check this prediction at a few test points by evaluating your expression from part (a) using values of d » L. For example, you might calculate the exact and approximate results using d=1.0 m and again using d =3 m, or any distances that are much larger than the length. Use at least two different test points so you can see that the result has the right dependence on the distance - if you triple the distance, as I did in my suggested distances, what should happen to the potential? (e) Should the potential at A be larger, smaller or equal to the potential at a test point B, a distance d to the right of the right end of the rod? Justify using a physics argument, then find an expression for the potential at B and check that the math answer agrees with your physics one. 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.)
A thin plastic rod of length L has a positive charge Q uniformly distributed along its length. *A d I d *B
Transcribed Image Text:A thin plastic rod of length L has a positive charge Q uniformly distributed along its length. *A d I d *B
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