Laplace Equation – Cartesian & Cylindrical Coordinates

1. Standard Laplace Equation (Cartesian Coordinates)

The Laplace equation describes steady-state fields such as electric potential, temperature, or fluid flow:

∇²V = 0

In Cartesian coordinates (x, y, z), it is written as:

∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0

• V → potential function
• ∂²V/∂x², ∂²V/∂y², ∂²V/∂z² → second derivatives in each direction

2. Laplace Equation in Cylindrical Coordinates

Cylindrical coordinates are (r, φ, z), where r = radial distance, φ = azimuthal angle, z = axial coordinate.

∇²V = (1/r) ∂/∂r ( r ∂V/∂r ) + (1/r²) ∂²V/∂φ² + ∂²V/∂z² = 0

3. Special Cases in Cylindrical Coordinates

• Radial only (V depends on r only):
  (1/r) d/dr ( r dV/dr ) = 0
• Axial only (V depends on z only):
  d²V/dz² = 0
• 2D cylindrical (no z dependence):
  (1/r) ∂/∂r ( r ∂V/∂r ) + (1/r²) ∂²V/∂φ² = 0

4. Applications

• Electric potential in Cartesian geometries (plates, boxes)
• Electric potential in cylindrical structures (wires, coaxial cables)
• Temperature distribution in rods and cylinders
• Magnetic field analysis in cylindrical systems

Exam Tip: Cartesian Laplace → ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0
Cylindrical Laplace → ∇²V = (1/r) ∂/∂r ( r ∂V/∂r ) + (1/r²) ∂²V/∂φ² + ∂²V/∂z²