(WILL RATE LIFESAVER!!!!) I need help with parts a through d, specifially part c

Question

(WILL RATE LIFESAVER!!!!)

I need help with parts a through d, specifially part c and d. thank you!!!!

Line of charge on the y-axis at P = (0, b). We'll repeat part of what I did lecture to set up problem 3. (a) Set-up. Draw a line of length L (draw it a long thin rectangle) along the y-axis centered at (0, 0). It should look vertical. We will consider the line chopped up in to N pieces. Furthermore we will approximate each little piece to be a point charge at the middle of the piece. For N = 3, draw the chopped up boundaries and the point charges. The location of each piece is (0, Y), where i = 0, 1, ... N - 1 labels the piece. So Y_i means the value on the y-axis for the i^th point. Write the width of the piece dy in terms of the symbolic parameters given so far. Write the expression for the chunk of charge dQ_i. It should contain some or all of the following terms: lambda, dy, and i. (b) Differential element. Consider a point P on the x-axis with coordinate (a, b). Write the expression for dE^x_i and dE^y_i. Express cos and sin in terms of the parameters such as x and a. (c) Analytic solution & limit. For a special point on the y-axis P = (0, b) where b > L/2, integrate the above expression to find the analytic form of E. (d) For b > > L, show that the equivalent system to produce the same E is a point charge at the origin with total charge Q = lambda L

Explanation / Answer

2. a. so for length L

divided into 3 parts

location of centres of three segmets are (0,L), (0,0), (0,-L)

where dy = L/3

charge on each chunk dq = lambda*dyi = lambda*dy

b. for the point on x axis , coordinates = (x,0)

so, dEix = k*xdq/(x^2 + yi^2)^3/2

dEiy = k*yi*dq/(x^2 + yi^2)^3/2

c. for a point on y axis, (0,b), b > L/2

dEx = 0

dEy = k*dq/(b - y)^2 = k*l;ambda*dy/(b - y)^2

integrating

Ey = k*lambda[1/(b - L/2) - 1/(b + L/2)] = k*lambda[4L/(4b^2 - L^2)]

d. for b > > L

Ey = k*lambda[1/(b - L/2) - 1/(b + L/2)] = k*lambda[4L/(4b^2 - L^2)]

4b^2 > > L^2

Ey = k*lambda[L/(b^2)]

Ey = kQ/b^2

where Q = lambda*L

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