A sphere of radius \( a \) has a bound charge \( Q \) uniformly distributed over its surface. The sphere is immersed in a uniform fluid dielectric medium with a fixed dielectric constant \( \epsilon \), as shown in Fig. Additionally, the fluid contains a free charge density given by:
\( \rho(r) = -kV(r) \),
where \( k \) is a constant, and \( V(r) \) represents the electric potential at a distance \( r \) from the center, relative to infinity.
Compute the electric potential \( V(r) \) everywhere, assuming \( V = 0 \) as \( r \to \infty \).
1
For k<0 , \( V(r) = \begin{cases} \frac{Q }{4\pi a (1 + \alpha a)}, & r \leq a \\ \frac{Q e^{-\alpha(r-a)}}{4\pi r (1 + \alpha a)}, & r > a \end{cases} \)
2
For k>0, \( V(r) = \begin{cases} \frac{Q }{4\pi a (1 + \alpha a)}, & r \leq a \\ \frac{Q e^{-\alpha(r-a)}}{4\pi r (1 + \alpha a)}, & r > a \end{cases} \)
3
For k<0 , \( V(r) = \frac{Q}{4\pi (\beta a \sin(\beta a) + \cos(\beta a))} \frac{ \cos(\beta a)}{r}, \, r < a \)
4
For k >0 ,\( V(r) = \frac{Q}{4\pi (\beta a \sin(\beta a) + \cos(\beta a))} \frac{ \cos(\beta a)}{a}, \, r \leq a \)