Volume Integration is one of the important mathematical tools that are used in Electromagnetics. This article discusses its definition and interpretation as in Electromagnetism.
Almost every Science/Engineering pupil is aware of the integration. 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.
An electromagnetic problem generally needs line Integration, Surface Integration, and Volume Integration.
Volume Integration in Electromagnetics
Consider any object having volume say for example a solid metal sphere. Let us assume that its density is ρ kg/m3 and volume is V m3. If we are asked to calculate the mass of this sphere, we can calculate quickly as-
Mass = Density × Volume = ρV Kg
Now let us have a twist. Assume that density ρ is given as the certain function of coordinates of the point which you are considering. This case wouldn’t be so easy as above and we cannot conclude our answer of mass merely by multiplying the density function with the volume. At every small point of the given volume, the density must be calculated at that point and multiplied with infinitesimal volume around that point.
In other words, to find the mass in the latter case, we have to use the integral calculus by finding the infinitesimal mass by considering the infinitesimal volume and collecting all such infinitesimal pieces all over the volume of sphere i.e. taking integration.
This type of integration is called volume integration.
Representation of the Volume Integration
Volume integration is represented as shown below-
where ρ is the scalar function distributed over the volume like density, volume charge density etc. and dv is the infinitesimal/ differential volume for the given ranges of variables.
Now as explained in line and surface integration, volume integration can be understood as:-
Calculating the infinitesimal product of the scalar function at a point and small (infinitesimal) volume around that point. Then summing up all such infinitesimal products covering all the volume.
Now we know that to have a volume or to cover the complete volume, all three coordinates must be covered. In other words, the integration would be the triple integration covering the range for each variable. so it is also represented as-
Differential volume (dv) in different Coordinate Systems
In every coordinate system, the small or differential volume is obtained by multiplying all three differential lengths.
Can volume integration be calculated for vector field?
Volume on its own is a scalar quantity and volume integrations are mainly calculated for scalar functions like density, volume charge density, temperature etc. But it can also be calculated for vector functions by considering each component separately.
More stuff:- YouTube Channel