I got the y component of the E-Field at point R but i am not sure where to start

Question

I got the y component of the E-Field at point R but i am not sure where to start on this part. If you could please show all steps and work so i can understand that would be great! I keep getting people copying and pasting answers from other websites that make no sense.

An infinitely long solid insulating cylinder of radius a = 4.1 cm is positioned with its symmetry axis along the z-axis as shown. The cylinder is uniformly charged with a charge density rho = 37.0 microC/m3. Concentric with the cylinder is a cylindrical conducting shell of inner radius b = 16.6 cm, and outer radius c = 19.6 cm. The conducting shell has a linear charge density lambda = -0.46microC/m.

What is V(c) - V(a), the potentital difference between the outer surface of the conductor and the outer surface of the insulator?

Explanation / Answer

Gauss's law inside the inductor:

E (2rL) = (a2L)/

E = (a2)/(2r) = 37e-6*0.041*0.041/(2*8.85e-12)/r = 3514/r

note that inside the inductor teh electrical field is zero or the potential diffrence between b and c is zero, therefore,

V(b) = V(c)

therefore

V(c) - V(a) = V(b) - V(a) = E.dr = (3514/r).dr = 3514 (lna - lnb)

>>>> V(c) - V(a) = 3514 (ln(0.041) - ln(0.166)) = -4914 V

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