A uniform line charge that has a linear charge density = 3.3 nC/m is on the x ax

Question

A uniform line charge that has a linear charge density = 3.3 nC/m is on the x axis between x = 0 to x = 5.0 m.

(a) What is its total charge?
.... nC

(b) Find the electric field on the x axis at x = 6 m.
.... N/C

(c) Find the electric field on the x axis at x = 10.0 m.
.... N/C

(d) Find the electric field on the x axis at x = 270 m.
.... N/C

(e) Estimate the electric field at x = 270 m, using the approximation that the charge is a point charge on the x axis at x = 2.5 m.
.... N/C

(f) Compare your result with the result calculated in part (d) by finding the ratio of the approximation to the exact result. To do this, you will need to assume that the values given in this problem statement are valid to more than two significant figures.

.....
(g) Is your approximate result greater or smaller than the exact result?

... greater

... smaller

Explanation / Answer

a) total charge = linear charge density x Length

     = 3.3 x 10^-9 x 5 =16.5 x 10^-9 C = 16.5 nC

b) considering a part of width dx at x,

dq = 3.3 x 10^-9 dx

dE = kdq / R^2

dE = (9 x 10^-9)dq / ( 6 -x)^2 = (9 x 3.3)dx / ( 6 -x )^2 = 29.7dx / (6 -x)^2

intergrating,

E = 29.7 / (6 -x)   x is from 0 to 5 m

E = (29.7 / 6) - (29.7 / 1) = 24.75 N/C

c) E = 29.7 / (10 - x)

x is from 0 to 5.
E = 2.97 N/C

d) E = 29.7 / (270 - x)

x is from 0 to 5.

E= 2.07547 x 10^-3 N/C

e) E = kq / r^2

r= 270 - 2.5 = 267.5 m

E = (9 x 10^-9 x 16.5 x 10^9 ) / 267.5 =2.07529 x 10^-3

f) Eexact / Eapprox = 1.0000985

g) approximate result is smaller.

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