Suppose that a magnetic field is produced in some region of

Suppose that a magnetic field is produced in some region of space by sources such as bar magnets, electric currents, etc. Select any two locations in this region (A and A\') and connect these two points by a broken line made of short displacements (s_i). An example of this is shown in figure 1. These segments are numbered 1 through 8 and in this particular example it takes all eight to move from A to A\'. As you move through the magnetic field the effect of the field adds up. For example, suppose the first displacement (s_1) had a magnitude of 0.5cm and pointed approximately 70 degree above the positive x-axis. The other s_i vectors would then have displacements described in similar fashions. In figure 2 a magnetic field (caused by distant sources) has been sketched. (The path of figure 1 is unchanged). There is an important quantity we can form called the line integral of the magnetic field along the given path from A to A\'. This line integral is defined as the sum of the dot products of each displacement vector s_j and the magnetic field vector B_j at the location of S_j. This line integral can be approximated by the sum of a finite number of dot products: beta m f = sigma_f B_j middot S_j

Solution

Line integral is basically summing up the values at consideration along a specific path.

As shown in the figure above, the line integral along path from A to A\' passing along the sections S1, S2, S3 and so on would be integral of the magnetic field at each of those points along the specific curves, all added up together.

Hence we can write: Line integral from A to A\' = B1. S1 + B2.S2 + B3.S3 and so on

That is the line integral = B_{j .} S_j

NOTE: Magnetic fields are produced by, say, electric currents. Also, we know that magnetic fields lines curl themselves around the source and have their strength proportional to the strength of the total current.

The line integral which we evaluate along a closed curve basically combines all the three points above to give an expression for the current lying inside the curve chosen. Let us assume that the current passing through an area is I_enc and we choose a curve around it represented by S, then the line integral for the magnetic field around the curve S gives us: B.dl = I_enc

The situation given in the problem mentioned above is an introductory to the approach we take for evaluation of such line integrals in a region filled with magnetic field lines.