My issue is getting to the answer below with the information derived within. I u

Question

My issue is getting to the answer below with the information derived within. I used the (kQz)/(z^2+R^2)^3/2 equation with the given values (.03m) and (9x10-6C). The answer I entered was me going down some different rabbit holes. Could you show me steps on how to get the correct answer shown?

A thin nonconducting rod with a uniform distribution of positive charge Q is bent into a circle of radius R. The central perpendicular axis through the ring is a z-axis, with the origin at the center of the ring (a) what is the magnitude of the electric field due to the rod at z = 0? 0 N/C (b) what is the magnitude of the electric field due to the rod at z ? 0 N/C (c) In terms of R, at what positive value of z is that magnitude maximum? 0.707 R (d) If R = 3.00 cm and Q = 9.00 pC, what is the maximum magnitude? 8.87e5X3.46e+07 N/C

Explanation / Answer

E=(kQz)/(z^2+R^2)^3/2

dE/dz =0

==>kQ[((z^2+R^2)^(3/2) -Z*(3/2)*2Z*(z^2+R^2)^(1/2)]/(z^2+R^2)^3 =0

==>kQ[((z^2+R^2)^(3/2) -3Z^2(z^2+R^2)^(1/2)]=0

==>kQ[((z^2+R^2)-3Z^2](z^2+R^2)^(1/2)=0

==>[((z^2+R^2)-3Z^2]=0

==>R^2 =2Z^2

==>Z=R/sqrt2

=>Z=((sqrt2)/2)R

=>Z=0.707R

MAXIMUM E=(kQz)/(z^2+R^2)^3/2

E=(kQ(1/(sqrt2))R)/((2/4)R^2+R^2)^3/2

E=(kQ(1/(sqrt2))R)/((3/2)R^2)^3/2

E=(kQ(1/(sqrt2))R)/[(3sqrt3/2sqrt2)*R^3]

E=(kQ(2/(3sqrt3))/[R^2]

E=(9*10^9)(9*10^-6)(2/(3sqrt3))/[0.03^2]

E=(81000)(2/(3sqrt3))*10000/[9]

E=(3000)(2)*10000/sqrt3

E=34641016

E=3.46*10^7 N/C

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