Line integral

Line Integral

The line integral  is much like the surface integral only the integration is along a line, not around a surface.

 

2 Important cases

 

 

1) If B is everywhere perpendicular to a line, the line integral of B is :

 

2) If B is everywhere tangent to a line of length L, and  has the same magnitude B at every point, the line integral of B is :

       

Line integral around a closed curve

 

Initial and final point of integration are the same:

Note circle indicating closed curve.

 

Integration of magnetic field  around a wire

 
B wire = (u0I)/2πd (you knew that!)
B is constant at distance d and everywhere tangent to the circle.
Mathematically speaking:

  

 Integration of magnetic field  around a wire

 

BL = 2πdB

Bwire = (u0I)/2πd

Therefore: