in this example we’re given that in a coil
of resistance r, magnetic flux due to an external
magnetic field, varies with time as, phi is
equal to k, c minus t square, weber. and we’re
given that k and c are positive constants.
we’re required to find the total heat produced
in coil in time from zero to c. in this situation
the very 1st thing we can do is, we can calculate
the e m f induced, in coil is, this e we can
write as, mod of, d phi by d t. on differentiating
this expression we get it as, 2 k t. now if
this is the e m f induced, as it is due to
the time varying magnetic field this’ll
be the loop e m f. and using this we can directly
calculate, the induced current in the coil,
and here induced current we can write as loop
e m f by the total resistance of coil, so
it’ll be 2 k t by, r. we’ve got the current
as a function of time, then we can easily
calculate the heat produced, in coil, by using
the expression h is integration of i square
r d t, from zero to c, which we already studied
in thermal effects of electric current. we
integrate it from zero to c. on putting the
value of current we can see we get integration
from zero to c. putting the value of current
it’ll be 4 k square t square by r d t. integrating
we get 4 k square by r is a constant, and
integration of t square will be t cube by
3 within limits from zero to c. on substituting
the value of upper limit and lower limit,
the result we get as 4 by 3, k square c cube
by r. that’ll be the answer to this problem
that is the total heat produced in the duration
from zero to c.
