(a) Most accurate approximation: E=[(Q/A)/(2e0)] * [1- (z/(R^2+z^2)^1/2] = _____

Question

(a) Most accurate approximation: E=[(Q/A)/(2e0)] * [1- (z/(R^2+z^2)^1/2] = ________ N/C (b) Less accurate approximation: E=[(Q/A)/(2e0)] * [1-(z/R)] = ________ N/C (c) Least accurate approximation: E=[(Q/A)/(2e0)] = ________ N/C (d) How good are the approximate formulas at this distance? answer _______________________________________ ______________________________________________ (e) At what distance does the least accurate approximation for the electric field give a result that is closest to the most accurate? (choose 1-4)       1. close to the disk       2. at a distance R       3. at a distance R/2       4. far from the disk Note: Sorry for the choppy equations, but the Add/Edit Custom Equation function on Cramster is not working properly on any computer i have tried it on.       

Explanation / Answer

Given radius of disk r = 21 cm charge on the disk q = 7*10^ -6 C distance between point and the center of the disk z = 3 cm a) electric field at [point p due to charged disk                   E=[(Q/A)/(2e0)] * [1- (z/(R^2+z^2)^1/2]          Area of disk A = pr^2 = 3.14 ( 21*10^ -2 )^2 = 0.1384 m                    E = [ 7*10^ -6 / 0.1384)/(2*8.85*10^ -12 ) ][1- 0.03 /(0.1915+9)^( 1 /2)                      =2.45*10^ 6 N /C b) E=[(Q/A)/(2e0)] * [1-(z/R)]            = [ 7*10^ -6 / 0.1384)/(2*8.85*10^ -12 ) ][1-  0.03/0.21]          = 2.4492*10^ 6 N /C c) E=[(Q/A)/(2e0)]           = [ 7*10^ -6 / 0.1384)/(2*8.85*10^ -12 ) ]            = 2.8575*10^ 6 N /C d)least accurate approximation for the electric field give a result that is closest to the most accurate is at far from the disk              [(Q/A)/(2e0)] * [1- (z/(R^2+z^2)^1/2]= [(Q/A)/(2e0)]                      [1- (z/(R^2+z^2)^1/2]= 1                  (R^2+z^2)^1/2 - z   = (R^2+z^2)^1/2             squaring on both sides                R^2 + z^2 + z^2 - 2 z . (R^2+z^2)^1/2 = R^2 + z^2                             z^2 - 2 z . (R^2+z^2)^1/2 = 0                              z^2  = 2 z . (R^2+z^2)^1/2                               z = 2 (R^2+z^2)^1/2                  squaring on both sides                       z^2 / 4   = R^2 + z^2                              - 3z^2 = R^2                              z = R / v3 so the answer is 4

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