Is the Electric Field Unique with Given Charge Density and Boundary Conditions?

  • Thread starter ehrenfest
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In summary, the problem states that the field is uniquely determined when the charge density rho is given and either V or the normal derivative of V is specified on each boundary surface. Knowing the normal derivative does not provide any additional help in solving the problem. Using the divergence theorem and Poisson's equation, we can find the integral of the field and the charge density, making the use of the curl, divergence, and gradient theorems unnecessary.
  • #1
ehrenfest
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[SOLVED] Griffiths Problem 3.4

Homework Statement


Prove that the field is uniquelly determined when the charge density rho is given and either V or the normal derivative [itex]\frac{\partial{V}}{\partial{n}}[/itex] is specified on each boundary surface. Do not assume the boundaries are conductors, or that V is constant over any given surface.


Homework Equations





The Attempt at a Solution


If you know V on the surface, this is the Corollary to the First Uniqueness Theorem. I don't see how knowing the normal derivative, [itex]\frac{\partial{V}}{\partial{n}}= \nabla{V}\cdot \hat{n}[/itex], helps at all.
 
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  • #2
My guess: probably one of those Stoke's theorems (curl, divergence, gradient) will simplify things.
 
  • #3
The divergence theorem and Poisson's equation tell us that [tex]\int_{S} \nabla V \cdot \hat{n} da = \int_{V}\nabla ^2V d\tau = \int_{V}\rho/\epsilon_0 d\tau [/tex]. We know rho, so we didn't even need the normal derivative to get that integral. Thus, I don't see how the curl, divergence, gradient theorems will help.
 
  • #4
anyone?
 
  • #5
Never mind. I got it.
 

Related to Is the Electric Field Unique with Given Charge Density and Boundary Conditions?

1. What is the problem statement for Griffiths Problem 3.4?

The problem states that a particle with mass m and charge q is placed in a uniform electric field E and a uniform magnetic field B. The particle is initially at rest and then released. The goal is to find the trajectory of the particle.

2. What are the key concepts needed to solve Griffiths Problem 3.4?

To solve this problem, one needs to have a good understanding of the equations of motion for a charged particle in an electric and magnetic field, as well as knowledge of vector calculus and basic kinematics.

3. What are the steps to solving Griffiths Problem 3.4?

The steps to solving this problem are to first write out the equations of motion for the particle, taking into account the electric and magnetic fields. Then, use vector calculus to simplify the equations and solve for the velocity and position of the particle. Finally, apply the initial conditions to find the trajectory of the particle.

4. What is the significance of solving Griffiths Problem 3.4?

Solving this problem allows us to better understand the behavior of charged particles in electric and magnetic fields, which has applications in various fields such as particle accelerators, magnetic resonance imaging, and plasma physics.

5. Can the solution to Griffiths Problem 3.4 be applied to real-world situations?

Yes, the solution to this problem can be applied to real-world situations where a charged particle is subjected to both an electric and magnetic field. It can also be used to understand the behavior of particles in experiments or in natural phenomena such as cosmic rays.

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