2.12:
Divergence and Curl
The divergence of a vector field at a point is the net outward flux per unit volume, as the volume at that point shrinks to zero.
Mathematically, divergence is the dot product of the del operator with the vector field.
Consider a vector field of water flowing through a pipe. If the water flows with constant velocity, it does not diverge.
On passing through a hole, the velocity of water diverges, resulting in a positive divergence.
On connecting the pipe to a multi-holed connector, the water velocity decreases, leading to negative divergence.
The curl of a vector field is the circulation of the vector per unit area, as this area shrinks to zero. It is directed normal to the area where the circulation is maximum. Mathematically, it is the cross product of del operator with the vector field.
Consider a non-uniform velocity vector of a river. A stick tossed into the river floats smoothly where the velocity vector has zero curl, and rotates where the velocity vector has non-zero curl.
2.12:
Divergence and Curl
The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the vector field is "contracting" or "converging" towards that point. This implies that the vector field is flowing inwards towards the point more than it is flowing outwards. This point is often called the "sink" of the field.
The divergence is zero if the inward flux at a point equals the outward flux. Mathematically, divergence is the dot product of the del operator with the vector field and is expressed as
The curl of a vector field is the circulation of the vector per unit area as this area tends to zero, and is in the direction normal to the area where the circulation is maximum. The curl of a vector field indicates the local rotation or circulation of the vector field calculated at any arbitrary point. A zero curl indicates no rotation, while a non-zero curl indicates rotation of the vector field. Mathematically, curl is the cross product of the del operator with the vector field and is expressed as
Curl is an important concept in many areas of physics, including electromagnetism and fluid dynamics. In electromagnetism, the curl of electric and magnetic fields determines the behavior of electromagnetic waves. Meanwhile, in fluid dynamics, the curl of the velocity field determines the degree to which a fluid "circulates" or "rotates" at a given point.
A curl indicates direction of a non-uniform flow, whereas divergence of the field only shows the scalar distribution of its sources.
Suggested Reading
- Griffiths, D.J. (2013). Fourth Edition.Introduction to ELectrodynamics. San Francisco, CA: Pearson. pp.17.
- Ida, N. (2015). Fourth Edition.Engineering Electromagnetics. Switzerland: Springer International Publishing. Pp. 67, 76.