Surface Integration in Electromagnetics

Surface Integration is one of the useful mathematical tools that are used in Electromagnetics. In this article let us overview the surface integration along with its interpretations.

The term integration is known to almost every Science/Engineering pupil. Formally in Mathematics, the integration is interpreted as the area under the given curve. For example, say we are given with the curve f(x). We define dx as an infinitesimal or differential length along the X-axis. For dx to be very small, the multiplication of f(x)dx, is considered as the area of the strip, as shown in the figure. By collecting the strips from x =a to x = b is termed as an integration.

There are many advanced concepts related to the integration in Mathematics. But from Electromagnetics point of view, we can summerise it as discussed above.

To solve Electromagnetic problems, generally, we need line Integration, Surface Integration, and Volume Integration.

Click here to read about Line Integration.

Consider any vector field A is present in the region. Let us say a surface of an area S is placed within the field as shown in the figure.

Now, the lines of the vector field will be passing through the surface. The total number of lines of the field that are passing through the given surface normally is termed as the flux of the field. And flux of the vector field through the given surface is represented by the surface integration.

Representation of the Surface Integration

Surface integration is represented as shown below: –

Where A is the vector field and ds is the infinitesimally small area from the given surface.

One point is worth noting here that in Electromagnetics, surface or plane is considered as a vector. For example, let us say we have a square of side 10 cm. Then its area will be 100 cm². So in the Electromagnetic Fields, this square will be represented by a vector of magnitude (length) 100 cm² and direction normal (perpendicular) to the plane.

As stated above the surface integration gives us the flux of the vector field through the given surface. But in the process of integration, we first consider an infinitesimally small area or differential area from the given surface. As shown in the above representation diagram as ds.

As discussed just above ds would be the vector. So by taking dot product with ds vector, we will get the component of the field along the ds vector [by the definition of dot product]. As ds vector is already normal (perpendicular) to the surface, in other words, we get component of the field normal to the surface.

Then we sum up this product by covering all the ds present on the surface. This is represented by the integration sign in the representation. Actually, integration is the operation in which we sum up the effects of all the differential elements. As we cover all the differential elements i.e. ds, we have two variables changing to cover the complete area/surface. That means, in other words, when we say the surface integration we inherently mean the double integration.

Recall from the Electromagnetics Basics Course for a surface or plane two coordinates are variable and rest is constant.

As surface integration require double integrations, it can also be represented as: –

Differential surface (ds) in different Coordinate Systems

We already know, in Electromagnetics, we use Rectangular, Cylindrical, and Spherical Coordinate Systems. So we must define differential element ds for each of them as shown below: –

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