Question

At what radius along the perpendicular bisector of a wire of length 0.29 m carrying 30 nC of charge does the assumption of cylindrical symmetry in applying Gauss's law give you an answer whose error exceeds 5%?

Answer 1

To apply Gauss's law w assume the charge distribution of very long wire and assume cylidrical sysmetry of the field around the wire

charge distribution per unit length   = 30/0.29 nC= 103.5 nC /m

field at a distance r from the wire

E(r) = /2 r

but the actual filed will not be uniform

To get the field on the perpendicular bisector.

we consider a small element of length dl on the charge

and the field dE at a perpendicular distance r from the wire

dE = dl/ 4 (r² +y²)

for every element on onside of the bisector thee is another element on the other side of it and all the components parallel to the wire get canceled and components perpendicular to it will add up the field due to the total wire on the perpendicular bisector. We consider the length of the wire as 2l and

= 30nC/2l

parallel component = dE Cos(p) = dE * r/( r²+y²)^1/2

, y is the distance of the element from the enter of the wire

we have to integrate it for y = -l to +l

integrating it we get

E'(r) = l/ 2 r(r² +l²)^1/2 , where l= 0.29/2 m

E(r) = E'(r) ( 1+l²/r²)^1/2  

E(r) exceeds E^'(r) by 5% when

( 1+l²/r²)^1/2 = 1.05

l²/r² = 0.1025

r = l/0.32 = 0.29/2(0.32) = 0.45

at 0.45 m on the perpendicular bisector the filed exceeds by 5% than the actual field by assuming Gauss law.