A uniform line charge of linear charge density = 3.2 nC/m extends from x = 0 to

Question

A uniform line charge of linear charge density = 3.2 nC/m extends from x = 0 to x = 5 m.

(a) What is the 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 = 11 m.

N/C

(d) Find the electric field on the x axis at x = 290 m.

N/C

(e) Find the field at x = 290 m, using the approximation that the charge is a point charge at the origin.

N/C

Compare your result with that for the exact calculation in part (d). (field for the point charge divided by the field for the line charge)

Please do not just put differentiations - I would like substitutions so I can follow, please.

Explanation / Answer

a. total charge = 3.2 *5 = 16 nC

b.

ets take a small length element dx, q is the total charge, l is total length, a is distance from nearer end.
dq=(q/l)dx

dE=dq/{4%u03C0%u03F5o(a+x)^2 }
substitute for dq:

dE = qdx / {4%u03C0%u03F5ol(a+x)^2}

E = q / 4%u03C0%u03F5ol%u222B_0^l%u2592(a+x) ^(-2)dx (integrating giving proper limits upper limit=l, lower limit= 0)

E=q / {4%u03C0%u03F5o(a+l)}

solve for q=3.5nC/m*5m=17.5nC, l = 5m, a = 6 - 5 = 1m

e) use columb's law directly, taking q=17.5nC, r= 270m,

rest you can do by simply following the process of answer b.

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