Circle with Nonuniform Charge

Determine the potential on the z-axis of a nonuniform charged circle that has a linear charge density

a) $lambda =C*sin phi$

b) $lambda =C*sin 2phi$

c)$lambda =C*sin 3phi$

d) $lambda =Cphi$

$d q=λ*d L=λR*dϕ$  and $d V(P)=(kλ*d L)/sqrt{(z^2+R^2 )}=(kλR*dϕ)/sqrt{(z^2+R^2 )}$

for $λ=C* sin⁡ ϕ$  we have

$V(P)=∫_0^2πkCR/sqrt{(z^2+R^2 )}*sin⁡ ϕ*dϕ=0$

for $λ=C*sin⁡ 2ϕ$ we have

$V(P)=k C R/sqrt{(z^2+R^2 )} ∫_0^2π sin⁡ 2ϕ*dϕ=-k C R/sqrt{(z^2+R^2 })*cos⁡ 2ϕ/2 |_0^2π=0$

for $λ=C*sin⁡ 3ϕ$ we have

$V(P)=k C R/sqrt{(z^2+R^2 )} ∫_0^2π sin⁡ 3ϕ*dϕ=-k C R/sqrt{(z^2+R^2 )}*cos ⁡3ϕ/3 |_0^2π=0$

for $λ=C*ϕ$ we have $V(P)=k C R/sqrt{(z^2+R^2 )} ∫_0^2π ϕdϕ=2π^2  k C R/sqrt{(z^2+R^2 )}$