Figure 3.15(a) shown two circular rings of radii `a` and `b` `(a gt b)` joined together with wires of negligible resistance. Figure 3.15(b) shown thew pattern obtained by folding the small loop in the plane of the large loop.
(a)
The pattern shown in Fig.3.15( c ) is obtained by twisting the small loop of Fig.3.15(a) through `180^(@)`.
All the three arrangement are placed in a uniform time varying magnetic field `dB//dt = k`, perpendicular to the plane of the loops. if the resistance per unit length of the wire is `lambda`, then determine the induced current in each case.

Explanation

(a) `phi_(B) = pi(a^(2) + b^(2))B`
`epsilon = (dphi)/(dt) = pi(a^(2) + b^(2)) xx (dB)/(dt) = pi(a^(2) + b^(2))k`
Induced current, `I = (epsilon)/(R ) = (pi(a^(2) + b^(2))k)/( lambda[2lambda(a + b)]) = (k(a^(2) + b^(2)))/(2lambda(a + b))`
(b) `phi_(B) = pia^(2)B - pib^(2)B = piB(a^(2) - b^(2))`
`epsilon = (dphi_(B))/(dt) = (pidB)/(dt) xx (a^(2) - b^(2)) = pik(a^(2) - b^(2))`
Induced current, `I = (epsilon)/(R ) = (pik(a^(2) - b^(2)))/(2pilambda(a + b)) = (k(a - b))/(2lambda)`



( c) Here induced emf in both loops will oppose each other, hence effective flux is given by
`Phi_(B) = pia^(2)B - pib^(2)B = piB(a^(2) - b^(2))`
`rarr` `epsilon = (dPhi_(B))/(dt) = pik(a^(2) - b^(2))`
induced current, `I = (epsilon)/(R) = (k(a - b))/(2lambda)`