Question

A circular conducting loop is held fixed in a uniform magnetic field that varies in time according to B(t) = B exp(-at) where t is in s, a is in s^-1 and B is the field strength in T at t = 0. At t = 0, the emf induced in the loop is 0.0614 V. At t = 2.74 s, the emf is 0.0197V. Find a.

Answer 1

Induced EMF is given as

E   =   -d?/dt

   here   ?   =   B* A   =   A * B₀ *e^-at

   =>   E   =   -d/dt (A * B₀ *e^-at)   =   a * A *B₀ * e^-at

      att   =   0,   E₀   =   0.0614 = a* A * B₀ * e^{-0 *}a  

         0.0614 =   a* A *B₀         -----------(1)

   at   t   = 2.74 s

      E_2.74 =    0.0197=  a* A * B₀ * e^{-a *2.74}

substitute the value of a * A* B₀   from equation (1)

   0.0197 =   0.0614* e ^{-2.74* a}

   e ^{2.74 *a}   =   0.0614/0.0197 = 3.1167

2.74 *a   =   ln (3.1167)  

   a   = 0.41488 s^-1