A point charge P is b = 9 m above the center of a uniformly charged ring of radi

Question

A point charge P is b = 9 m above the center of a uniformly charged ring of radius a = 0.14 m sitting in the xy plane with charge Q = 4 ?C.
Randomized Variables

b = 9 m
Q = 4 ?C
a = 0.14 m

(a) Calculate the z component of the electric field at P in newtons per coulomb.
(b) Calculate the x and y components of the electric field at P in newtons per coulomb.

(c) Assume b is very large. Now if b doubles, what happens to the magnitude of the Electric Field?

(please give me all the steps and answer)

Explanation / Answer

for any infinitisimal charge dq, making an angle with Z axis theta , the electric field's z component

is given by k*dq*cos(theta)/(b^2+a^2)

cos(theta)=b/sqrt(b^2+a^2)

substituting, we get

electric field component=k*b*dq/(b^2+a^2)^1.5

integrating from q=0 to q=Q, we get

electric field component=k*b*Q/(b^2+a^2)^1.5

putting all the values, we get electric field's z component=444.28 N/C

b) as the ring of charge is symmetrical, all other components of electric field on the axis will be 0.

hence x component=0

y component=0

c)if b is very large, the electric field becomes k*b*Q/b^3=k*Q/b^2

now if b doubles, the electric field strength will become a quarter of previous value.

hence 5th option is correct.

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