EXAMPLES OF SOLUTION OF LAPLACE’s EQUATION NAME: Akshay kiran E.NO.: SUBJECT: EEM GUIDED BY: PROF. SHAILESH SIR

Capacitance and Laplace’s Equation  Capacitance Definition  Simple Capacitance Examples  Laplace and Poison’s Equation  Laplace’s Equation Examples  Laplace’s Equation - Separation of variables  Poisson’s Equation Example

Potential of various charge arrangements 

Basic Capacitance Definition A simple capacitor consists of two oppositely charged conductors surrounded by a uniform dielectric. An increase in Q by some factor results in an increase D (and E) by same factor. With the potential difference between conductors: Q -Q E, D S B A.. increasing by the same factor -- so the ratio Q to V 0 is constant. We define the capacitance of the structure as the ratio of stored charge to applied voltage, or Units are Coul/V or Farads

Laplace and Poisson’s Equation 1. Assert the obvious  Laplace - Flux must have zero divergence in empty space, consistent with geometry (rectangular, cylindrical, spherical)  Poisson - Flux divergence must be related to free charge density 2. This provides general form of potential and field with unknown integration constants. 3. Fit boundary conditions to find integration constants.

Derivation of Poisson’s and Laplace’s Equations These equations allow one to find the potential field in a region, in which values of potential or electric field are known at its boundaries. Start with Maxwell’s first equation: where and so that or finally:

Poisson’s and Laplace’s Equations (continued) Recall the divergence as expressed in rectangular coordinates: …and the gradient: then: It is known as the Laplacian operator.

Summary of Poisson’s and Laplace’s Equations we already have: which becomes: This is Poisson’s equation, as stated in rectangular coordinates. In the event that there is zero volume charge density, the right-hand-side becomes zero, and we obtain Laplace’s equation :

Laplacian Operator in Three Coordinate Systems (Laplace’s equation)

Example 1 - Parallel Plate Capacitor d 0 x V = V 0 V = 0 Plate separation d smaller than plate dimensions. Thus V varies only with x. Laplace’s equation is: Integrate once: Integrate again Boundary conditions: 1. V = 0 at x = 0 2. V = V 0 at x = d where A and B are integration constants evaluated according to boundary conditions. Get general expression for potential function

Parallel Plate Capacitor II General expression: Boundary condition 1: 0 = A(0) + B Boundary condition 2: V 0 = Ad Finally: d 0 x V = V 0 V = 0 Boundary conditions: 1. V = 0 at x = 0 2. V = V 0 at x = d Equipotential Surfaces Apply boundary conditions

Parallel Plate Capacitor III Potential Electric Field Displacement d 0 x V = V 0 V = 0 Equipotential Surfaces E Surface Area = S n At the lower plate n = a x Conductor boundary condition Total charge on lower plate capacitance Getting 1) Electric field, 2) Displacement, 3) Charge density, 4) Capacitance

Example 2 - Coaxial Transmission Line V0V0 E L V = 0 Boundary conditions: 1.V = 0 at  b 2.V = V 0 at  a V varies with radius only, Laplace’s equation is: (  0) Integrate once: Integrate again: Get general expression for potential

Coaxial Transmission Line II V0V0 E L V = 0 Boundary conditions: 1.V = 0 at  b 2.V = V 0 at  a General Expression Boundary condition 1: Boundary condition 2: Combining: Apply boundary conditions

Coaxial Transmission Line III V0V0 E L V = 0 Potential: Electric Field: Charge density on inner conductor: Total charge on inner conductor: Capacitance: Getting 1) Electric field, 2) Displacement, 3) Charge density, 4) Capacitance

Example 3 - Concentric Sphere Geometry a b V0V0 E V = 0 Boundary Conditions: 1.V = 0 at r = b 2.V = V 0 at r = a V varies only with radius. Laplace’s equation: or: Integrate once: Integrate again: Boundary condition 1: Boundary condition 2: Potential: Get general expression, apply boundary conditions

Concentric Sphere Geometry II a b V0V0 E V = 0 Potential: (a < r < b) Electric field: Charge density on inner conductor: Total charge on inner conductor: Capacitance: Get 1) electric field, 2) displacement, 3) charge density, 4) capacitance