A 250 -nC point charge is placed at the center of an uncharged spherical conducting shell \(20 \mathrm{cm}\) in radius. Find (a) the surface charge density on the outer surface of the shell and (b) the electric field strength at the shell's outer surface.

Electric field strength is a way of quantifying the intensity of the electric field at a given point in space. It can be thought of as the force exerted per unit charge at that particular location. Imagining it like a wind blowing against a sail helps visualize how an electric charge would be 'pushed' by this field. The formula to determine the electric field strength due to a point charge is \(E = \frac{Q}{4\pi\epsilon_0r^2}\), where \(E\) represents the electric field strength, \(Q\) is the charge causing the electric field, \(\epsilon_0\) is the permittivity of free space, and \(r\) is the distance from the charge to the point where you're measuring the field.

For a point charge, this strength diminishes with the square of the distance from the charge, which means that as you move away from the charge, the 'strength of the wind' decreases rapidly. This inverse-square law is a fundamental principle governing electrostatic interactions. Knowing the field strength is crucial to understanding how charges interact with each other and can provide insights into the behavior of charged particles in various physical contexts.

A spherical conducting shell serves a special role in electrostatics. When a charge is placed inside such a shell, it induces a separation of charges within the conductive material. Charges in the conductor rearrange themselves so that the electric field within the bulk of the conductor is zero. This phenomenon is a consequence of the conductive properties of the shell's material, which allows free electrons to move in response to the internal electric field.

From the outside, the shell appears to be uniformly charged, and the behavior of the electric field outside the shell is as if all the induced charge were situated at the center. This result is known as the shell theorem, and it's a key element in solving many electrostatic problems. In the context of the exercise, this means that when calculating the surface charge density or the electric field at the shell's outer surface, we can treat the entire charged shell as if it were a point charge located at its center.

Permittivity of free space, often denoted as \(\epsilon_0\), is a physical constant that is key in determining the strength of the electric field due to a given charge in a vacuum. The value of \(\epsilon_0\) is approximately \(8.85 \times 10^{-12} C^2/N\cdot m^2\) and it forms part of Coulomb's law which governs the interactions between electric charges.

The permittivity of free space essentially tells us how much electric field (force per unit charge) will be produced per unit charge in a vacuum or, put simply, it's a measure of how easily an electric field can permeate through the vacuum of space. In the context of our exercise with the spherical conducting shell, it's a vital part of the equation used to calculate the electric field strength on the shell's surface. Understanding this concept not only applies to the immediate problem but also broadens comprehension of electric field interactions in various environments, whether they be in vacuum, air, or other media with different permittivity values.