Consider two concentric spherical conducting shells centered at the or...
Given:
- Two concentric spherical conducting shells centered at the origin.
- Outer radius of the inner shell is r_{a}.
- Inner radius of the outer shell is r_{b}.
- Charge density rho = 0 in the region r_{a} < r="" />< />
- V = 0 at r = r_{a}.
- V = V_{0} at r = r_{h}.
To find:
- V in the region r_{a} < r="" />< />
Explanation:
1. Electric Potential:
The electric potential at a point in space is given by the equation:
V = k * Q / r
Where:
- V is the electric potential.
- k is the electrostatic constant.
- Q is the charge enclosed.
- r is the distance from the center of the charge.
2. Applying Gauss's Law:
According to Gauss's law, the electric field inside a conductor is zero. This means that the electric potential is constant throughout the conductor.
3. Electric Potential on the Inner Shell:
Since the charge density rho = 0 in the region r_{a} < r="" />< r_{b},="" there="" is="" no="" charge="" enclosed="" within="" the="" inner="" shell.="" therefore,="" the="" electric="" potential="" on="" the="" inner="" shell="" is="" />
4. Electric Potential on the Outer Shell:
The charge enclosed within the outer shell is the difference between the charges enclosed by the outer shell and the inner shell. Since the charge density is zero in the region r_{a} < r="" />< r_{b},="" the="" charge="" enclosed="" by="" the="" outer="" shell="" is="" zero.="" therefore,="" the="" electric="" potential="" on="" the="" outer="" shell="" is="" also="" />
5. Electric Potential in the Region r_{a} < r="" />< />
Since the electric potential is constant throughout a conductor, the electric potential in the region r_{a} < r="" />< r_{b}="" is="" also="" />
Final Answer:
The electric potential in the region r_{a} < r="" />< r_{b}="" is="" zero.="" r_{b}="" is="" />