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A few examples of problems are given from electromagnetic theory and other areas of physics are given in which the line integral appears.
1. Types of Line integrals in Physics
The line integrals appear in many applications such computing
work done in mechanics, definition of e.m.f. and circulation of magnetic field in Ampere's law in electromagnetic theory etc.
Examples of simplest line integral is length of a curve between two points. Another example is that of moment of inertia of a rod about a fixed axis. The another kind of line integrals is work done in force field. For this type of examples, curve comes with an orientation. Yet another kind of line integral appears in computation of electric field of charge distributed continuously along a curve. This is simply a set of three integrals each being a weighted average of a line element.
More complicated examples of line integrals are of type \(\int \vec{a}\times \vec{dl}\)
2. Line integrals in mechanics
- Distance traveled
- Weight of a rod
- Moment of inertia of a rod
- Center of mass of a rod, ring etc.
3. Line integrals in electromagnetic theory
- Electric field of line charge
- Electric potential of line charge, of a ring.
- Work done in an external force field.
- Path independence of work in electrostatics.
- Ampere's law and circulation of magnetic field.
- EMF induced in a circuit.
