Q: How is an electric dipole field described mathematically?

An electric dipole (charges +q and −q separated by distance d) produces a field that falls off as 1/r³ at large distances — faster than the monopole 1/r² decay. The field in polar coordinates is E_r=2p cosθ/(4πε₀r³), E_θ=p sinθ/(4πε₀r³) where p=qd is the dipole moment. The same form describes magnetic dipoles (bar magnets, current loops) and gravitational quadrupole fields.

Question 1

What are divergence and curl of a vector field?

Accepted Answer

Divergence ∇⋅F measures how much a vector field spreads out from a point — positive for a source (e.g., electric field near a positive charge), negative for a sink. Curl ∇×F measures the rotational tendency; the z-component is ∂Fy/∂x − ∂Fx/∂y. Irrotational flow has ∇×v=0; incompressible flow has ∇⋅v=0. These operators appear throughout fluid mechanics, electromagnetism, and heat transfer.

Question 2

What is a streamline?

Accepted Answer

A streamline is a curve that is everywhere tangent to the velocity vector in a steady flow. Fluid particles travel along streamlines, and streamlines never cross (except at stagnation points). Numerically, streamlines are computed by integrating dx/ds=Fx/|F|, dy/ds=Fy/|F| using Euler or Runge-Kutta methods. In potential flow they are orthogonal to equipotential lines.

Question 3

What is the Rankine vortex?

Accepted Answer

The Rankine vortex combines a rigid-body rotation core (v∝r for r<R) with a free vortex exterior (v∝1/r for r≥R), closely matching real tornado and hurricane profiles. The core eliminates the infinite velocity at r=0 of the pure free vortex. It is widely used as an initial condition or analytic reference solution in computational fluid dynamics.

Question 4

How is an electric dipole field described mathematically?

Accepted Answer

An electric dipole (charges +q and −q separated by distance d) produces a field that falls off as 1/r³ at large distances — faster than the monopole 1/r² decay. The field in polar coordinates is E_r=2p cosθ/(4πε₀r³), E_θ=p sinθ/(4πε₀r³) where p=qd is the dipole moment. The same form describes magnetic dipoles (bar magnets, current loops) and gravitational quadrupole fields.

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