Curl formula explained in detail with an intuitive proof.

The curl of the vector field is one of the basic operations that are used in the study of Electromagnetics. This article explains the derivation for the Curl formula in brief.

What is the Curl of a vector field?

The Curl is defined as the 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 can be considered analogues to the rotation of the given vector field around the unit area. More is the field circulating along the unit area at the given point, more will be the curl. And the direction of the curl vector can be thought of the axis of the rotation of that imaginary unit area according to right-hand rule. Check out the following article for the definition in detail.

What is the Curl of a vector field?

The definition cum formula for the Curl

As discussed in the article mentioned above, the definition of the Curl is like this –

Definition of the Curl of a vector field

This illustrative formula is practically of no use while solving the numerical. Because we are supposed to find out the curl of the vector field whose expression is generally given. So let us elaborate this equation to extract the more useful curl formula.

The derivation of the Curl formula

Few Assumptions

The curl, in simple words, is the rotating or whirling nature of the vector field at a given point. Again in simpler words, it can be explained as – Assume that I have put a small surface at that point in the hypothetical vector field similar to force. Let us also assume that the surface has a fixed centre but flexible axis. Then, this surface can rotate about the particular axis. The position of the axis and magnitude at that point for the maximum rotation can be considered as the Curl of this vector field.

As I have assumed the vector field is similar to force, it would be able to rotate that imaginary surface. But actually, for any other vector field, we would say the circulation of the field lines around that surface. Isn’t it? More circulation more curl. And as I explained in the last article, the numerator of the above formula denotes the circulation.

Assume that any vector field E is present in the region and we are finding the curl at any point within the field say P(x₀, y₀, z₀).

Now according to the definition, the field lines of E will be circulating along the differential surface present at point P. And the axis of the curl will be decided using right-hand rule so that maximum circulation of the field is possible at that position. This position or orientation of the axis can be anywhere in the space. But for simplicity consider the axis along positive X-axis. Actually, the complete curl effect will be the combined effect (vector addition) of all three axes together i.e. along X, along Y and along Z.

As I am assuming the curl-axis along +X axis, I have to show the differential or small surface dS in YZ plane (bounded by ABCDA) and the circulation of the field in anticlockwise direction as shown in the figure below. This is according to the right-hand rule where thumb denoting the axis which is perpendicular to the surface and fingers denoting circulation of the field.

I am assuming the differential surface dS i.e. infinitesimally small surface bounded by differential lengths. Hence the line integration at the given point can be thought as the multiplication of the field value i.e. E and the differential length at that point.

The Intuitive derivation for the Curl formula

Let me again present the definition of the curl.

Let us calculate the close line integration in the numerator. As our small surface is bounded by four lengths viz. AB, BC, CD and DA. So this close line integration is calculated as –

Consider the very first integration i.e. from A to B. Let me present the diagram once again.

The length AB is the infinitesimally small length along Y-axis hence it can be considered as ‘dy’. The value of the function E at the location AB can be approximated using Taylor’s series. In the article named “The formulas of the Divergence with an intuitive explanation!“, I have explained how any function can be expanded around the given point in terms of spatial derivatives. In this case, the y-component of the field E can be expressed with Taylor’s series expansion around P(x₀, y₀, z₀) as shown below. I am considering only y component of the vector field E i.e. E_y for a reason. Can you guess why?

Curl formula Derivation Steps - Taylor's series

I have considered only E_y because I am considering the integration from A to B. So for this integration, I need the component of the field E along AB i.e. along Y-axis, hence only E_y . Now the given line AB is at distance (dz/2) below from the point P. Hence from the above formula, we can consider only the term consisting derivative w.r.t. z. Also, z is taken equal to [z₀ – (dz/2)]. So the value of E_y at the location of the line AB is as follows –

So the integration of E along AB can be written for the differential case as follows. The E_y is given by the above equation and the dl is dy as stated initially.

You can easily work out in a similar manner to get the remaining integrals for BC, CD and DA. In each case, the value of the field E and dl are modified accordingly.

So for the complete close integration along ABCD i.e. circulation of the field, we need to add all these four terms together and we will get the following.

But, wait. Can you observe the term dydz. What does this represent? It is the area of the differential surface dS that we have assumed at the starting of our discussion. Isn’t it? Also as we have assumed this area to be very small, differentially small we can say it is approaching zero. So above relation can be written as follows –

Bingo! Check out the left-hand side of the above equation. It is exactly the same as the definition of the Curl. Am I right? So this is the expression of the curl of our assumed vector field E having axis along X-axis (as we have assumed so initially).

On similar lines, we can proceed step by step as we did here and find the Y and Z components of the Curl.

Rather than these three formulas for different components, the complete Curl formula in matrix form is represented as follows. This matrix form formula after simplification would also lead to the above three formulas.