Laplace Equation

The Laplace Equation is a second-order partial differential equation and it is denoted by the divergence symbol ▽. It is a useful approach to the determination of the electric potentials in free space or region. The Laplace equation is derived to make the calculations in Physics easier and it is named after the physicist Pierre-Simon Laplace. In this article, we will learn ‘What is Laplace equation Formula’, solving Laplace equations, and other related topics. 

Laplace Equation Formula

Laplace equation used in Physics is one of the first applications of these equations. The Laplace equation formula was first found in electrostatics, where the electric potential V, is related to the electric field by the equation E=−▽V, this relation between the electrostatic potential and the electric field is a direct outcome of Gauss's law, ▽.E = ⍴/ε₀, in the free space or in other words in the absence of a total charge density. The equation is also seen while studying the gravitational fields, where the gravitational potential V, related to the gravitational field by g=−▽V. The Laplace equation has wide applications and it is being used whenever we encounter potential fields. 

Not only in electrostatics the Laplace equation is found to be used in the various branches of Physics, such as in thermal Physics, where the potential V will be replaced by the temperature (it implies that, the Laplace equation will be written in the form of temperature gradient), and in fluid mechanics, the potential V will be replaced by the velocity field of an incompressible fluid (it implies that the Laplace equation will be written in the form of a gradient of velocity field), etc. Laplace's equation formula plays a vital role in most of the branches of Physics and Mathematics.

Laplace's equation in Physics is an illustration of a partial differential equation, which involves a number of independent variables. In general, the potential V is independent of the variables x,y, and z and the differential equation must be integrated to explain the simultaneous dependence of the potential V on these three variables. This can be a really tedious problem, one that is quite harder to solve than any ordinary differential equation involving an independent variable. 

The Laplace equation in Physics consists of two important properties. The first property states that the solution of the Laplace equation formula is unique once when solved under a suitable number of boundary conditions employed. The second property states that the solutions of the Laplace equation formula hold good with the superposition principle.

Laplace Equation Derivation

Now, let us have a look at the formulation of the Laplace equation and Laplace equation derivation. The formulation of Laplace's equation includes any number of boundaries, on which the potential V is particularly defined. Examples of these formulations are familiarly known as boundary-value problems, and these are found mostly in electrostatics. 

For instance, we have a peculiar boundary-value problem usually asked for the potential V and between conductors, on which the potential V is constant. In such cases, the surface of each conductor is considered as a boundary, and by knowing the constant value of the potential V on each boundary, we can determine a unique solution to Laplace's equation in the space between the conductors. 

In other cases, the boundary can be different than the conducting surface, and the potential V might be varying on the boundary. Yet the property remains unchanged, that once the potential of the conductor is specified on each boundary, the solution to Laplace's equation formula between boundaries will be a unique solution. This is known as the uniqueness theorem, and it states that if we can determine a solution that will satisfy Laplace's equation and the boundary condition V=V0 on the conducting surface, then the obtained solution will be a unique solution of Laplace’s Equation.

The superposition principle arises directly from the fact that Laplace's equation is continuous in the potential V. Suppose that V1, V2, V3, and so on, are all the solutions to Laplace's equation so that ▽²V_j=0. Any superposition of the form

V=a₁V₁+a₂V₂+a₃V₃+.......+a_j V_j…….(1)

where aj are constants, is also a solution, because

⇒▽²V=▽²(a1V1 + a2V2 + a3V3 +....) =a₁▽V₁+a₂▽V₂+a₃ ▽V₃+...= 0……(2)

▽²V=0….(3)
Equation (3) is known as the Laplacian equation and ▽² is known as the Laplacian operator.

Equation (2) is the statement of the superposition principle, and it will form an integral part of our approach to find the unique solution to Laplace's equation with proper boundary conditions. It is important for one to understand that the superposition principle applies to any number of solutions Vj, this number could be finite or infinite depending upon the number of variables included.

This mathematical operation is obtained in equation (2), the divergence of the gradient of a potential V is called the Laplacian equation. In general, the Laplacian equation can be defined as the divergence of the gradient of any function. The function might be changing depending upon the concept of interest. Expressing the Laplacian equation in different coordinate systems (cartesian coordinate system, spherical coordinate system, and cylindrical coordinate system) to take advantage of the symmetry of a charge configuration assists in the solution for the electric potential V. For example, if the charge distribution has spherical symmetry, then the Laplacian equation will be expressed in terms of the polar coordinates.

Since the electric potential is a scalar function, this method has advantages over trying to determine the electric field directly. Once the electric potential has been estimated, the electric field can be calculated by considering the gradient of the electric potential i.e., E=▽Vj.

Now let us have a look at the different forms of Laplace equation examples in Physics. 

1. ▽²V=0, The Laplace equation electrostatics defined for electric potential V.

2. If g =-▽V then▽²v=0, the Laplace equation in gravitational field.

3. ▽²u=0,u is the velocity of the steady flow.

In general, the Laplace equation can be written as ▽²f=0,where f is any scalar function with multiple variables.

Applications of Laplace Equation

• Anything which is directly related to a linear differential equation can be easily solved by using the Laplace equation. Differential equations occurred mainly in Physics, Mathematics, and engineering.

• The Laplace equations are used to describe the steady-state conduction heat transfer without any heat sources or sinks.

• Laplace equations can be used to determine the potential at any point between two surfaces when the potential of both surfaces is known.

• The capacitance between the two surfaces can be found using Laplace’s and Poisson’s equation.
  
  

Solved Examples:

Let V = 4x²yz³ at a given point P (1,2,1), then find the potential V at P and also verify whether the potential V satisfies the Laplace equation.

Sol:

Given,

The potential V= 4 x² y z³and we are asked to determine the potential V at point P (1, 2, 1). 

The potential V at point P is given by:

⇒ V=4 (12) (2) (13)

⇒ VP=8 volts

Now, let us verify the Laplace equation for the potential V at point P. The Laplace equation is given by:

⇒▽²V=0

Since we are calculating in cartesian coordinates we get:

⇒▽²V= ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²=0

⇒ ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²=0

⇒ 8yz³ + 0 + 24x²yz = 0

⇒16 + 480 ≠ 0

Therefore, the potential V at point P is 8 volts and it does not satisfy the Laplace equation.