To solve this problem, we need to understand the behavior of the electric field due to a uniformly charged hollow metal sphere.$\\$Consider a hollow metal sphere of radius $R$ with a uniform charge distribution.$\\$The electric field $E$ at a distance $r$ from the center of the sphere is given by:$\\$• For $r < R: \\$Inside a hollow conductor, the electric field is zero. This is due to the electrostatic shielding effect.$\\ E = 0 \\$• For $r \geq R: \\$The electric field behaves as if all the charge were concentrated at the center of the sphere.$\\ E = \frac{kQ}{r^2} \\$where $k$ is Coulomb's constant and $Q$ is the total charge on the sphere.$\\$Now, let's analyze the options:$\\$Option 1: Increases as $r$ increases for $r < R$ and for $r > R \\$This is incorrect because the electric field is zero for $r < R$ and decreases for $r > R. \\$Option 2: Zero as $r$ increases for $r < R,$ decreases as $r$ increases for $r > R \\$This is correct because the electric field is zero inside the sphere and decreases with $r^2$ outside.$\\$Option 3: Zero as $r$ increases for $r < R,$ increases as $r$ increases for $r > R \\$This is incorrect because the electric field decreases with $r^2$ for $r > R. \\$Option 4: Decreases as $r$ increases for $r < R$ and for $r > R \\$This is incorrect because the electric field is zero for $r < R. \\$Therefore, the correct option is Option 2.