An electrical current is going through a circular coil. The coil is input partially in a region with magnetic field.
1) Calculate the electromagnetic force which is acting on coil.
2) If the coil would be input totally in the magnetic field, which is the elastic tension from the coil?
1) We'll take an element of coil, ds=R*d tetha. The elementary force is acting on the element, this force being radially oriented:
dF=I*dr x B, module(dr)=ds, where
r and B are vectors
For each and every element ds, does exist an element, symmetrically from the axis of symmetry OX, so that the components perpendicular to the ox axis are cancelling each other. We'll consider only ox components:
F= Integral (dF* cos tetha)=Integral (I*ds*B*cos tetha)=
=Integral( IRBcos tetha d tetha)=2IRB Integral(cos tetha*dtetha)=2IRB*sin tetha
For a semi-coil, tetha=pi/2, F=2IRB
2) The result of the two tensions which are acting at the ends of the element ds, must balance the electromagnetic force dF, which is acting on the power element.
2T sin (dtetha/2)=2T(d tetha/2)=T d tetha
sin x tends towards to x, when x tends towards to 0.
T d tetha=dF=IdsB=IR(d tetha)B