Some queries on uniqueness theorem

In summary, when a charge is placed inside a cavity of a solid conductor, the induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way according to the Uniqueness Theorem of Electromagnetism. This is because the electric field and surface charges are determined by the electrostatic potential, which is a unique solution of Poisson's equation given the known charge inside the cavity and the boundary conditions.
  • #1
VishweshM
2
0
Consider a solid conductor with a cavity inside. Place a charge well inside the cavity. The induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way. How does this follow from the Uniqueness Theorem of EM? David Griffith claims this but never gets around to explain it in his book. Any thoughts?
 
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  • #2
All densities of charge are determined by the normal component of the electric field near the surface:
[tex]
\sigma = \epsilon_0 E_n
[/tex]

The electric field in turn is determined by the electrostatic potential [itex]\varphi(\mathbf x)[/itex]. This potential is a solution of Poisson's equation
[tex]
\Delta \varphi = -\frac{\rho}{\epsilon_0},
[/tex]
given the known charge inside the cavity and the boundary condition that potential is constant throughout the metal.

The uniqueness theorem states that the solution of this equation and conditions is unique. Since the potential determines everything, the surface charges are unique too.
 

Related to Some queries on uniqueness theorem

1. What is the uniqueness theorem?

The uniqueness theorem is a mathematical concept that states that if a mathematical object or system satisfies certain conditions, then it is unique and there can be no other object or system that satisfies the same conditions.

2. How is the uniqueness theorem used in science?

In science, the uniqueness theorem is used to prove the uniqueness of solutions to certain mathematical equations or physical systems. This allows scientists to make predictions and draw conclusions about the behavior of these systems.

3. What are the conditions that must be satisfied for the uniqueness theorem to apply?

The conditions for the uniqueness theorem vary depending on the specific situation, but generally, the system or object must be well-defined and the equations or laws governing it must be consistent and well-behaved.

4. Can the uniqueness theorem be applied to real-world phenomena?

Yes, the uniqueness theorem can be applied to real-world phenomena, such as the behavior of physical systems or the solutions to certain mathematical equations. It is a fundamental concept in many scientific fields.

5. Are there any limitations to the uniqueness theorem?

While the uniqueness theorem is a powerful tool in many scientific disciplines, it does have some limitations. It may not apply in situations where the system or object is not well-defined, or when the equations or laws governing it are not consistent or well-behaved.

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