3. (3.5 pts) A thin annular disk lies in the xy plane, is centered on the...

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3. (3.5 pts) A thin annular disk lies in the xy plane, is centered on the origin, and has inner radius a and outer radius 2a as shown in the figure to the right. The disk has a charge density (s) 2 (9) , = 00 where s is in standard cylindrical coordinates, the distance from the z axis. P a. Show that the algebraic expression for the electric potential for any point P on the z axis V(0,0, z) = V(z) is V(z) = 6a (2z (z + a V (z') = xx - z + 4a) - a (z + a 4z + 4a (2z (z + a z + 4a - oa (22'2 (z + 1 - 2^2 + 4) - (z + 1 - 4z + 4) 6 Where the second expression is in terms of the unitless rescaled parameter z' = z/a. b. Plot the "normalized unitless" potential as a function of z' (using plot software). Include both sides of the origin in your plot. c. Where is the maximum values of the potential in part b? Explain why this makes physical sense. d. Determine the algebraic expression for the electric field on the z axis (z). (Hint: you already know V(z)). Using Mathematica and the "Simplify" function is often (but not always...) useful. e. Plot the normalized unitless electric field as a function of z' (using plot software). f. Qualititively, where is the maximum or maxima of the magnitude of the electric field? Explain 1) how this agrees with the plot in part 3b, and 2) how this makes physical sense. 3. (3.5 pts) A thin annular disk lies in the xy plane, is centered on the origin, and has inner radius a and outer radius 2a as shown in the figure to the right. The disk has a charge density (s) 2 (9) , = 00 where s is in standard cylindrical coordinates, the distance from the z axis. P a. Show that the algebraic expression for the electric potential for any point P on the z axis V(0,0, z) = V(z) is V(z) = 6a (2z (z + a V (z') = xx - z + 4a) - a (z + a 4z + 4a (2z (z + a z + 4a - oa (22'2 (z + 1 - 2^2 + 4) - (z + 1 - 4z + 4) 6 Where the second expression is in terms of the unitless rescaled parameter z' = z/a. b. Plot the "normalized unitless" potential as a function of z' (using plot software). Include both sides of the origin in your plot. c. Where is the maximum values of the potential in part b? Explain why this makes physical sense. d. Determine the algebraic expression for the electric field on the z axis (z). (Hint: you already know V(z)). Using Mathematica and the "Simplify" function is often (but not always...) useful. e. Plot the normalized unitless electric field as a function of z' (using plot software). f. Qualititively, where is the maximum or maxima of the magnitude of the electric field? Explain 1) how this agrees with the plot in part 3b, and 2) how this makes physical sense.