Three identical wires are bent into semi-circular arcs to each of radius `R`. These arcs are connected with each other to form a closed mesh such that one of tham lies in `x-y` plane, one in `y-z` plane and the other in `z-x` plane as shown in. In the region of space, a uniform magnetic field of induction `vec(B) = B_(0)(hat(i) + hat(j))` exists, whose magnitude increases at a constant rate `dB//dt = alpha`. Calculate the magnitude of emf induced in the mesh and mark the direction of flow of induced current in the mesh.

Answer

`2R6(2)(1 + (pi)/(2))alpha`; `Q `rarr` R `rarr` P `rarr` Q`

Explanation

`phi = vec(B).vec(A) = B_(0)(hat(i) + hat(j) + hat(k))`
where `A'` is area of one segment.
`=(1)/(2)(sqrt(2)Rsqrt(2)R) + (piR^(2))/(2) = R^(2) (1 + (pi)/(2))`
`:.` `phi = B_(0)A' + B_(0)A' = 2B_(0)A'`

`(dphi)/(dt) = |epsilon| = 2A'(db)/(dt)`
`epsilon = 2R^(2)(1+(pi)/(2))alpha`
The current flows `Q to R to P to Q`
`Q rarr R rarr P rarr Q`