Laplacian of a Scalar Field - Definition, Significance & Solved Examples.


- The laplacian is a scalar operator. Hence when applied to scalar field, the result is also a scalar field.



- Laplace operator is a second order differential equation defined as the divergence of the gradient of scalar field.


- Laplacian of a scalar field V is written as ∇2 V.


- Laplacian of a scalar is:

            - Used in electrostatics to represent the charge associated to a given potential.
            - Used in defining the Helmholtz equation of propagation of EM wave.
            - Used in Laplace’s and Poisson’s equation.


Laplacian of a scalar field V in Cartesian coordinate system is given as:






Laplacian of a scalar field V in Cylindrical coordinate system is given as:






Laplacian of a scalar field V in Spherical coordinate system is given as:






Q.1 Determine the Laplacian of the scalar fields:
a) V = x2y + xyz
Ans:





=> ∇2 V = [(d / dx) ( 2xy + yz)] +[ (d / dy) ( x2 + xz)] + [(d / dz) (xy)] = 2y


b) V = ρz sinφ + z2cos2 φ + ρ2
Ans:






= (1 / ρ) (z sinφ + 4ρ) – (1 / ρ2) (2ρ sinφ + 2z2 cos 2φ) + 2 cos2 φ

= 4 + 2 cos2 φ - (1 / ρ2) 2z2 cos 2φ



ALSO READ:

- Line , Surface and Volume Intergral.

- Del Operator - Definition and Significance.

- Gradient Of a Scalar (∇ V).

- Numericals / Solved Examples - Gradient Of a Scalar.

- Divergence Of a Vector ( ∇ . A ).

- Numericals / Solved Examples - Divergence Of a Vector.

- Curl Of a Vector ( ∇ x A).

- Laplacian Of a Scalar ( ∇2 V).


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