We need to determine the electric field at a point originating from a uniformly charged spherical shell. This problem requires applying Gauss's Law, a fundamental principle in electrostatics.
1. Gauss's Law states that the electric flux through any closed surface is directly proportional to the net charge enclosed within that surface:\[\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\]Where: - \( \vec{E} \) represents the electric field, - \( d\vec{A} \) denotes an infinitesimal area element, - \( Q_{\text{enc}} \) is the total charge enclosed, - \( \epsilon_0 \) is the permittivity of free space.
2. Due to the symmetrical charge distribution of a spherical shell, the electric field inside the shell is uniform and radially symmetric. - At any location within the spherical shell, the electric field contributions from all segments of the shell cancel out due to this symmetry. - Specifically, the field generated by a charge element on one side of the shell is nullified by the field from an opposing charge element on the other side. Consequently, the electric field inside the shell is zero.
3. It follows that the electric field at any point inside the shell is zero. This outcome is independent of the shell's radius or the total charge it contains.
4. Outside the shell (for \( r<R \)), the electric field behaves as if all the charge were concentrated at the shell's center. However, the question specifically asks for the field inside the shell. Therefore, options (1), (3), and (4) are incorrect.
Option (2) is correct as it accurately states that the electric field inside the shell is zero.