Chapter 7 - Poisson's and Laplace's Equations

# Chapter 7 - Poisson's and Laplace's Equations - Chapter 7...

This preview shows pages 1–6. Sign up to view the full content.

Chapter 7 Poisson’s and Laplace’s Equations

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Poisson's and Laplace’s Equations Point form of Gauss's Law: ∇ • D = ρ v Relationship between D and E : D = ε E Gradient relationship between V and E : E = - V Combining the three equations: ∇• D = ∇ • ( ε E ) = - ∇ • ( ε∇ V) = ρ v ∇ • ∇ V = Poisson’s Equation ε ρ - v Recall:
Note: ∇ • ∇ is abbreviated as 2 ("del squared") If the charge density is zero, then ∇ • ∇ V = 2 V = 0 The equation is now called Laplace's equation . The 2 operation is called the Laplacian of V .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Any expression for potential V is valid if it satisfied Laplace’s equation (when ρ s = 0) or Poisson’s equation (when ρ s 0) Laplace’s and Poisson’s equations relate the potential field to the charge density in a region.
In cartesian coordinates: ε ρ - = + + = v 2 2 2 2 2 2 2 z V y V x V V Cylindrical coordinates: 2 2 2 2 2 2 z V V 1 V 1 V + φ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 19

Chapter 7 - Poisson's and Laplace's Equations - Chapter 7...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online