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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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FAQs on PPT: Poisson’s & Laplace Equations - Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)
| 1. What is Poisson's equation and how is it used in physics? |  |
Ans. Poisson's equation is a partial differential equation that relates the distribution of a scalar field to its sources in physics. It is commonly used to describe phenomena such as the electric potential, gravitational potential, or fluid flow. By solving Poisson's equation, we can determine the potential or field distribution caused by the given sources.
| 2. What is Laplace's equation and why is it important in electrostatics? | |
Ans. Laplace's equation is a second-order partial differential equation that describes the behavior of scalar fields in physics. In electrostatics, Laplace's equation is used to model the electric potential in regions where there are no charges or external sources. Solving Laplace's equation allows us to determine the electric potential in these regions, which is crucial for understanding the behavior of electric fields and charges.
| 3. How are Poisson's and Laplace's equations related? | |
Ans. Poisson's equation is derived from Laplace's equation by introducing a source term. While Laplace's equation describes the behavior of scalar fields in regions without sources, Poisson's equation accounts for the presence of sources. Thus, Poisson's equation can be seen as an extension of Laplace's equation that incorporates the effect of sources on the field distribution.
| 4. Can Poisson's and Laplace's equations be solved analytically? | |
Ans. Yes, Poisson's and Laplace's equations can be solved analytically in certain cases. If the boundary conditions and sources are simple enough, it is possible to find analytical solutions using techniques such as separation of variables or Fourier series. However, in more complex situations, numerical methods like finite difference or finite element methods are often employed to approximate the solutions.
| 5. What are some applications of Poisson's and Laplace's equations in engineering and science? | |
Ans. Poisson's and Laplace's equations have numerous applications in various fields of engineering and science. Some examples include the analysis of electric and magnetic fields, fluid dynamics, heat conduction, and quantum mechanics. These equations are fundamental tools for understanding and solving a wide range of physical problems, making them essential in fields such as electrical engineering, physics, and computational science.
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