To determine the electric field at the center of the spherical cavity within a thick shell, we can apply Gauss's law and the principles of electrostatics. Let's break down the problem step by step to find the solution.
Understanding the Setup
We have a thick spherical shell with an inner radius of R and an outer radius of 3R, and it possesses a uniform charge density of σ (in coulombs per cubic meter). Inside this shell, there is a spherical cavity with a radius of R. Our goal is to find the electric field at the center of this cavity.
Applying Gauss's Law
Gauss's law states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Mathematically, it can be expressed as:
Φ = ∮ E · dA = Q_enc / ε₀
Where:
- Φ is the electric flux through the surface.
- E is the electric field.
- dA is the differential area element.
- Q_enc is the charge enclosed by the surface.
- ε₀ is the permittivity of free space.
Analyzing the Electric Field Inside the Cavity
To find the electric field at the center of the cavity, we can consider a Gaussian surface that is a sphere of radius r (where r < R) centered at the center of the cavity. Since the cavity is empty and there is no charge enclosed within this Gaussian surface, we have:
Q_enc = 0
According to Gauss's law, if there is no charge enclosed, the electric field inside the cavity must be:
E = 0
Why is the Electric Field Zero?
This result can be understood intuitively. The thick shell creates an electric field due to its charge distribution, but the cavity is a region where there is no charge. The electric field contributions from the charged shell surrounding the cavity effectively cancel out at the center of the cavity. This is a consequence of the symmetry of the charge distribution. In essence, for any point inside the cavity, the electric field due to the shell's charge is equal in magnitude but opposite in direction from all other points, leading to a net electric field of zero.
Final Thoughts
In summary, the electric field at the center of the spherical cavity within the thick shell is zero. This result highlights the importance of symmetry in electrostatics and the application of Gauss's law in determining electric fields in complex charge distributions.
