{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec14-1 - Poisson equation The Poisson equation can be...

This preview shows pages 1–3. Sign up to view the full content.

Poisson equation The Poisson equation can be written, 2 u ( ~ r ) = ρ ( ~ r ) For example, in two dimensions, u ( x , y ) and ρ ( x , y ) 2 x 2 + 2 y 2 u ( x , y ) = ρ ( x , y ) For the case of electrostatics, using Gaussian units we have 2 φ ( ~ r ) = - 4 πρ ( ~ r ) For a point charge, 2 φ ( ~ r ) = - 4 πδ ( ~ r - ~ r 0 ), and we find φ ( ~ r ) = 1 | ~ r - ~ r 0 | The function G ( ~ r ,~ r 0 ) = 1 | ~ r - ~ r 0 | is the Green function for the Poisson equation

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

View Full Document
Green function for the Poisson equation The Green function G ( ~ r ,~ r 0 ) = 1 | ~ r - ~ r 0 | solves the Poisson equation for a point source, 2 G ( ~ r ,~ r 0 ) = - 4 πδ ( ~ r - ~ r 0 ) Now we notice that ρ ( ~ r ) = R R R ρ ( ~ r 0 ) δ ( ~ r - ~ r 0 ) d τ , and then with the volume integral extending over the region of nonzero charge density,
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

lec14-1 - Poisson equation The Poisson equation can be...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online