in fig. The four rods have `lambda` resistance per unit length. The arrengement is kept in a magnetic field of constant magnitude `B` and directed perpendicular to the plane of the figure and directing in ward. Initially, the sides as shown form a square. Now each wire starts moving with constant velocity `v` toward the opposite wire.
Find as a function of time:
(a) induced emf in the circuit.
(b) induced current in the circuit with direction.
( c ) force required on each wire to keep its velocity consatnt.
(d) total power required to maintain constant velocity.
(e) thermal power developed in the circuit.

Explanation

(a) `E_(net) = 4Bl'v`

Equivalent circuit:
`:.` `E_(net) = 4B(l - 2vt)v`
`E_(net) = 4Bv(l - 2vt)`
(b) `I = (E_(net)) = (4r) = (4Bl'v)/(4lambdal')` `rarr` `I = (Bv)/(lambda)`
( c) Force required on each wire `= Il'B`
`Force = (B^(2)v)/(lambda)(l - 2vt)`
(d) Total power required to maintain constant velocity
`= 4Fv = (4B^(2)v^(2))/(lambda)(l - 2vt)`
(e) Thermal power developed in the circuit
`= 4I^(2)r = 4I^(2)lambda(l - 2vt) = 4(B^(2)v^(2))/(lambda^(2))lambda(l - 2vt)`
Thermal power `= (4B^(2)v^(2))/(lambda)(l - 2vt)`