Sunday, April 18, 2010

Faraday's law method

At any position of the loop the magnetic flux through the loop is

\Phi_B = \pm \int_0^{\ell} dy \int_{x_C-w/2}^{x_C+w/2} B(x) dx
= \pm \ell \int_{x_C-w/2}^{x_C+w/2} B(x) dx \ .

The sign choice is decided by whether the normal to the surface points in the same direction as B, or in the opposite direction. If we take the normal to the surface as pointing in the same direction as the B-field of the induced current, this sign is negative. The time derivative of the flux is then (using the chain rule of differentiation or the general form of Leibniz rule for differentiation of an integral):

\frac {d \Phi_B} {dt} =  (-) \frac {d}{dx_C} \left[ \int_0^{\ell}dy \ \int_{x_C-w/2}^{x_C+w/2} dx B(x)\right] \frac {dx_C}{dt} \ ,
  = (-)  v\ell  [ B(x_C+w/2) - B(x_C-w/2)] \ ,

(where v = dxC / dt is the rate of motion of the loop in the x-direction ) leading to:

 \mathcal{E} = -\frac {d\Phi_B} {dt} = v\ell  [ B(x_C+w/2) - B(x_C-w/2)] \ ,

as before.

The equivalence of these two approaches is general and, depending on the example, one or the other method may prove more practical.

0 comments:

Post a Comment