The figure (Figure 1) shows a thin rod of length L with total charge Q Find an e

Question

The figure (Figure 1) shows a thin rod of length L with total charge Q Find an expression for the electric field strength on the axis of the rod at distance r from the center Express your answer in terms of the variables L, Q, r, and appropriate constants Verify that your expression has the expected behavior if r > > L Express your answer in terms of variables Q, r and constants pi elementof_0. Evaluate E at r = 46 cm it L = 50 cm and Q = 33 nC Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

given a rod of length L with total cvharge Q

given a rod of length L with total cvharge Q

given a rod of length L with total cvharge Q

a. For a point P on its axis, at distance r from the center of the rod consider a very small part of rod of width dx at distance x fropm the center of the rod
   charge on this part, dq = Q*dx/L
   distance of this part from point P, d = r + x
   then electric field at point P due to this charge, dE = k*dq/d^2 = k*Q*dx/L*(r + x)^2
   so integrating from x = L/2 to x = -L/2 we get
   E = k*Q * [ 1/(r - L/2) - 1/(r + L/2) ] /L
   E = k*Q * [ r + L/2 - r + L/2 ] /L*(r - L/2) *(r + L/2)
   E = k*Q /(r - L/2) *(r + L/2) = 4kQ/(4r^2 - L^2)
B. if r >> L
   then 4r^2 - L^2 = 4r^2
   E = k*Q /(r - L/2) *(r + L/2) = 4kQ/(4r^2 - L^2) = 4kQ/4r^2 = kQ/r^2
C. at r = 4.6 cm = 0.046 m
   L = 5 cm = 0.05 m
   Q = 3.3 nC = 3.3*10^-9 C
   k = 8.98*10^9
   so E = 4*8.98*10^9*3.3*10^(-9)/(4*0.046^2 - 0.05^2) = 19875.251 v/M

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