Shows a square frame of wire having a total resistance r placed coplanarly with a long, straight wire. The wire carries a current I given by `i= i_0 sin omega t`. Find(a) the flux of the magnetic field through the square frame, (b) the emf induced in the frame and (c) the heat developed in the frame in the time interval 0 to `(20pi)/(omega)`.

Explanation

Considering an element dx at a distance x from
`we have (a) phi=B.A.`
`d(phi)=(mu_0 ixxadx)/_2 pi x)`
`phi= int d phi`
`=(mu_0 ia)/(2 pi) In[1+(a/b)]`
`(b) e=(d phi)/(dt)`
`=(d)/(dt) (mu_0 ia)/(2 pi) In[1+(a/b)]`
`(mu_0a)/(2 pi) In[1+(a/b)](d)/(dt) (i_0 sin omega t)`
` (mu_0 ai_0 omega cos omega t)/(2 pi r) In[1+(a/b)]`
`(c) i=(e/r)`
` (mu_0 ai_0 omega cos omega t)/(2 pi r) In[1+(a/b)]`
`H=i^2rt`
`=`[(mu_0 ai_0 omega cos omega t)/(2 pi r) In[1+(a/b)]^2xxr xxt`
`=(mu_0)^2 xxi^2xx omega^2)/(4 pi xx r^2) In^2[1=(a/b)]xxrxx(20 pi)/(omega)]`
`(5(mu_0)^2(a^2)(i_0)^2 omega)/(2 pi r)(In)^2[(1+(a/b)][:. t=(20 pi)/(omega).`