Page 1 Poisson’s and Laplace Equations A useful approach to the calculation of electric potentials Relates potential to the charge density. The electric field is related to the charge density by the divergence relationship The electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation In a charge-free region of space, this becomes Laplace's equation Page 2 Poisson’s and Laplace Equations A useful approach to the calculation of electric potentials Relates potential to the charge density. The electric field is related to the charge density by the divergence relationship The electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation In a charge-free region of space, this becomes Laplace's equation Potential of a Uniform Sphere of Charge outside inside Page 3 Poisson’s and Laplace Equations A useful approach to the calculation of electric potentials Relates potential to the charge density. The electric field is related to the charge density by the divergence relationship The electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation In a charge-free region of space, this becomes Laplace's equation Potential of a Uniform Sphere of Charge outside inside Poisson’s and Laplace Equations Poisson’s Equation From the point form of Gaus's Law Del_dot_ D r v Definition D D eE and the gradient relationship E DelV - Del_D Del_ eE () Del_dot_ eDelV ( ) - r v Del_DelV r v - e Laplace’s Equation if r v 0 Del_dot_D r v Del_Del Laplacian The divergence of the gradient of a scalar function is called the Laplacian. Page 4 Poisson’s and Laplace Equations A useful approach to the calculation of electric potentials Relates potential to the charge density. The electric field is related to the charge density by the divergence relationship The electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation In a charge-free region of space, this becomes Laplace's equation Potential of a Uniform Sphere of Charge outside inside Poisson’s and Laplace Equations Poisson’s Equation From the point form of Gaus's Law Del_dot_ D r v Definition D D eE and the gradient relationship E DelV - Del_D Del_ eE () Del_dot_ eDelV ( ) - r v Del_DelV r v - e Laplace’s Equation if r v 0 Del_dot_D r v Del_Del Laplacian The divergence of the gradient of a scalar function is called the Laplacian. LapR xx Vxy , z , () d d æ ç è ö ÷ ø d d yy Vxy , z , () d d æ ç è ö ÷ ø d d + zz Vxy , z , () d d æ ç è ö ÷ ø d d + é ê ë ù ú û := LapC 1 r r r r Vrf , z , ( ) d d × æ ç è ö ÷ ø d d × 1 r 2 ff Vrf , z , ( ) d d æ ç è ö ÷ ø d d é ê ë ù ú û × + zz Vrf , z , ( ) d d æ ç è ö ÷ ø d d + := LapS 1 r 2 r r 2 r Vr q ,f , ( ) d d × æ ç è ö ÷ ø d d × é ê ë ù ú û 1 r 2 sin q () × q sin q () q Vr q ,f , ( ) d d × æ ç è ö ÷ ø d d × + 1 r 2 sin q () 2 × ff Vr q ,f , ( ) d d d d × + := Poisson’s and Laplace Equations Page 5 Poisson’s and Laplace Equations A useful approach to the calculation of electric potentials Relates potential to the charge density. The electric field is related to the charge density by the divergence relationship The electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation In a charge-free region of space, this becomes Laplace's equation Potential of a Uniform Sphere of Charge outside inside Poisson’s and Laplace Equations Poisson’s Equation From the point form of Gaus's Law Del_dot_ D r v Definition D D eE and the gradient relationship E DelV - Del_D Del_ eE () Del_dot_ eDelV ( ) - r v Del_DelV r v - e Laplace’s Equation if r v 0 Del_dot_D r v Del_Del Laplacian The divergence of the gradient of a scalar function is called the Laplacian. LapR xx Vxy , z , () d d æ ç è ö ÷ ø d d yy Vxy , z , () d d æ ç è ö ÷ ø d d + zz Vxy , z , () d d æ ç è ö ÷ ø d d + é ê ë ù ú û := LapC 1 r r r r Vrf , z , ( ) d d × æ ç è ö ÷ ø d d × 1 r 2 ff Vrf , z , ( ) d d æ ç è ö ÷ ø d d é ê ë ù ú û × + zz Vrf , z , ( ) d d æ ç è ö ÷ ø d d + := LapS 1 r 2 r r 2 r Vr q ,f , ( ) d d × æ ç è ö ÷ ø d d × é ê ë ù ú û 1 r 2 sin q () × q sin q () q Vr q ,f , ( ) d d × æ ç è ö ÷ ø d d × + 1 r 2 sin q () 2 × ff Vr q ,f , ( ) d d d d × + := Poisson’s and Laplace Equations Given Vxy , z , ( ) 4y × z × x 2 1 + := x y z æ ç ç è ö ÷ ÷ ø 1 2 3 æ ç ç è ö ÷ ÷ ø := eo 8.85410 12 - × := Vxy , z , ( ) 12 = Find: V @ and r v at P LapR xx Vxy , z , ( ) d d æ ç è ö ÷ ø d d yy Vxy , z , ( ) d d æ ç è ö ÷ ø d d + zz Vxy , z , ( ) d d æ ç è ö ÷ ø d d + é ê ë ù ú û := LapR 12 = rv LapR eo × := rv 1.062 10 10 - ´ = Examples of the Solution of Laplace’s Equation D7.1Read More

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29 videos|49 docs|70 tests

### Introduction to Laplace and Poisson Equations

- Video | 16:45 min
### Solutions to Laplace’s Equations (Part - 1)

- Doc | 8 pages
### Solutions to Laplace’s Equations (Part - 2)

- Doc | 7 pages
### Solutions to Laplace’s Equations (Part - 3)

- Doc | 13 pages
### Solutions of Laplace Equation

- Video | 51:57 min
### Special Techniques (Part - 1)

- Doc | 10 pages

- Poisson’s and Laplace’s Equations
- Doc | 11 pages
- Electrostatics Potential & Capacitance
- Video | 18:01 min