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Put $\mu =\cos \vartheta$ and separate $\Phi =R(r)Y(\mu )$. Multiplying Laplace's equation by $r^2/(RY)$ givesFor the regular polynomial angular modes, $\lambda =n(n+1)$ and $Y=P_n(\mu )$ is a Legendre polynomial. The radial equation has branches $r^n$ and $r^{-n-1}$. Regularity at the origin keeps only the first inside, and decay at infinity keeps only the second outside. Continuity at $r=1$ identifies their coefficients, so a mode hasIts outward-minus-inward radial derivative is $-(2n+1)a_n{P}_n(\mu )$. The prescribed angular dependence decomposes as $V{\mu }^2=V{P}_0/3+2V{P}_2/3$, since $P_0=1$ and $P_2=(3{\mu }^2-1)/2$. Hence $a_0=-V/3$ and $a_2=-2V/15$, with no other required modes. The spherical harmonic matching with a derivative jump givesBoth branches are harmonic in their respective regions. They agree at the sphere, and their derivative jump is $V/3+(2V/3)P_2=V{\cos }^2\vartheta$. The interior is regular at zero and the exterior decays at infinity. An additional homogeneous matching mode would satisfy $-(2n+1)a_n=0$, so its coefficient vanishes.

Resolve the angular surface charge density into Legendre polynomials:For a single degree $\ell$, write the common surface potential as $V_{\ell }{P}_{\ell }$. Regularity and decay require radial factors $(r/R{)}^{\ell }$ inside and $(R/r{)}^{\ell +1}$ outside. Their exterior-minus-interior radial derivative at $R$ is $-(2\ell +1)V_{\ell }{P}_{\ell }/R$. Thus spherical harmonic matching with a derivative jump gives $V_{\ell }=R{\sigma }_{\ell }/(2\ell +1)$.

Only degrees zero and two occur, yieldingBoth expressions have surface value $2AR/3-2AR{P}_2/15$, and their derivative difference is $-2A/3+(2A/3)P_2=-A{\sin }^2\theta$, checking both interface conditions. The interior quadratic is regular at the origin, and the exterior potential tends to zero.

The leading far-field term is the monopole term. Indeed the total charge is $Q=R^2\int \sigma d\Omega =8\pi A{R}^2/3$, and the normalization of the specified jump gives $Q/(4\pi r)=2A{R}^2/(3r)$. No extra permittivity factor is inserted into the supplied normalization. Regularity here means finite piecewise smooth potential; its normal derivative necessarily jumps at the charged surface.