A uniform line charge of linear charge density lambda = 4.0 nC/m extends from x

Question

A uniform line charge of linear charge density lambda = 4.0 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 = 250 m. N/C (e) Find the field at x = 250 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)

Explanation / Answer

part a;

total charge=charge density*length=4 nC/m * 5 m=20 nC

part b:

consider a small strip of length dx at a location of x.

charge on this strip=dq=lambda*dx

electric field at x=6 m due to dq=dE=k*dq/(6-x)^2

where k=coloumb's constant=9*10^9

dE=k*lambda*dx/(6-x)^2

let 6-x=p

==>dx=-dp

dE=-k*lambda*dp/p^2

integrating both sides,

E=k*lambda/p

using p=6-x

E=k*lambda/(6-x)

using limits of x from x=0 to x=5:

electric field at x=6 m=9*10^9*4*10^(-9)*((1/(6-5))-(1/(6-0)))=30 N/C

part c:

consider a small strip of length dx at a location of x.

charge on this strip=dq=lambda*dx

electric field at x=11 m due to dq=dE=k*dq/(11-x)^2

where k=coloumb's constant=9*10^9

dE=k*lambda*dx/(11-x)^2

let 11-x=p

==>dx=-dp

dE=-k*lambda*dp/p^2

integrating both sides,

E=k*lambda/p

using p=11-x

E=k*lambda/(11-x)

using limits of x from x=0 to x=5:

electric field at x=11 m=9*10^9*4*10^(-9)*((1/(11-5))-(1/(11-0)))=2.7272 N/C

part d:

consider a small strip of length dx at a location of x.

charge on this strip=dq=lambda*dx

electric field at x=250 m due to dq=dE=k*dq/(250-x)^2

where k=coloumb's constant=9*10^9

dE=k*lambda*dx/(250-x)^2

let 250-x=p

==>dx=-dp

dE=-k*lambda*dp/p^2

integrating both sides,

E=k*lambda/p

using p=250-x

E=k*lambda/(250-x)

using limits of x from x=0 to x=5:

electric field at x=250 m=9*10^9*4*10^(-9)*((1/(250-5))-(1/(250-0)))=2.9387*10^(-3) N/C

part e:

if it is a point charge of 20 nC at origin, electric field at a distance of 250 m=9*10^9*20*10^(-9)/250^2=2.88*10^(-3) N/C

part f:
field for point charge/field for line charge=2.88/2.9387=0.98

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