Curl of a Vector field | The definition explained in detail.

Curl operation on the vector fields is often necessary for the study of Electromagnetics to find the circulation of the given field along a certain path. This article elaborates the basic definition.

A formal definition of the Curl

It is a vector whose magnitude is the maximum circulation of the given field per unit area (tending to zero) and whose direction is normal to the area when it is oriented for maximum circulation.

The Curl in simple words

In simple words, the curl can be considered analogues to the circulation or whirling of the given vector field around the unit area. More are the field lines circulating along the unit area around the point, more will be the magnitude of the curl.

The direction of the curl vector gives us an idea of the nature of rotation. It always follows the right-hand thumb rule where the thumb denotes the direction of curl vector and finger denotes the way of maximum circulation of the unit area.

The Curl – Explained in detail

The curl of a vector field is the mathematical operation whose answer gives us an idea about the circulation of that field at a given point. In other words, it indicates the rotational ability of the vector field at that particular point.

Technically, it is a vector whose magnitude is the maximum circulation of the given field per unit area (tending to zero) and whose direction is normal to the area when it is oriented for maximum circulation.

First of all, let me explain what do we mean by the circulation of the field. Let us consider any vector field is present in the region and let us also assume that a line XY is present in the field as shown in the figure below.

Now if we want to find the product of the component of the field along the line at every point and length of the line then we take line integral i.e.

In simple words, the line integration would give us the effect of the vector field along the given line. You can read more details of the line integration from the following article-

What is Line Integration in Electromagnetics?

Now let us consider the same vector field. But now consider the small area say ‘ds‘ bounded by the closed path (L) is present within the field.

Circulation of the vector field for the curl.

Now if I calculate the line integration of the given field along the path L, then in simple words, I would get the effect of the vector field along the L or boundary of the surface ‘ds‘. Now, what does this indicate? It is indicating how much the field is circulating the given area ‘ds‘. Isn’t it? It is represented as follows-

So this close line integration of the field around the boundary of the surface ‘ds’ is called as the circulation of the vector field. In layman’s words, it indicates the rotating or whirling capacity of the field if the surface is allowed to rotate. More is the circulation, more would be the answer of this integration.

Now again jump to the definition of the curl. It requires maximum circulation per unit area i.e. area should approach to zero. So, mathematically it can be written as follows

Now the final part of the definition that is the direction of the curl vector. According to the definition, it is normal to the area/surface such that the surface is aligned for the maximum possible circulation. Now, this can be easily determined using the right-hand thumb rule where thumb denoting the axis of the rotation if the surface is allowed to rotate according to the circulation of the field. And this will be the direction of the curl vector.

So mathematically, the definition would be as shown in the following figure where the bracketed term is the maximum circulation as discussed above and the unit vector according to the right-hand rule.