Question

A current / flows in the closed loop shown below as indicated. In general terms, find the magnitude and direction of the magnetic field at the center of the loop via Bio-Savart law. The circular arcs have the same center and all linear segments are radial.

Answer 1

linear segments do not have any magnetic field at the center of the loop as they pass through the center.

for any arc extending an angle of t, magnetic field at the center of the arc is given by

mu*I*t/(4*pi*r)

where mu=magnetic permeability=4*pi*10^(-7)

I=current in the arc

r=radius of the arc

direction of the magnetic field can be found out by right hand thumb rule

for both the arcs, direction of magnetic field will be in same direction

i.e. out of the plane of the paper

magnitude of magnetic field due to shorter arc=mu*I*theta/(4*pi*a)

magnitude of magnetic field due to longer arc=mu*I*(2*theta)/(4*pi*1.5*a)

=mu*I*theta/(3*pi*a)

hence total magnetic field =(mu*I*theta/(4*pi*a))+(mu*I*theta/(3*pi*a))

=7*mu*I*theta/(12*pi*a)