(Solved): An uncharged metal sphere of radius \( R \) is placed in an otherwise uniform electric field \( \d ...

An uncharged metal sphere of radius \( R \) is placed in an otherwise uniform electric field \( \downarrow \) \( \vec{E}=E_{0} \hat{z} \). Find the potential in the region outside the sphere \( \downarrow \) \( \downarrow \) Note: This problem corresponds to the following boundary (i) \( V=0 \) at \( r=R \downarrow \) (ii) \( V=-\vec{E}_{0} \cdot \vec{r}=-E_{0} r \cos \theta \) as \( r \rightarrow \infty \). \( \downarrow \) (Explanation: The external electric field will push positive charges to the "northern" surface of the metal, and negative charges to the "southern" surface. These induced charges, ir turn, distorts the field in the neighborhood of the sphere. Fas from the metal, the effects of these induced charge are negligible, and the potential will be that due to the uniform field only, i.e., as in (ii).) \( \downarrow \)