A wire is bent in the form of a square of side `'a'` in a varying magnetic field `bar(B) = alpha B_(0) t hat(k)`. If the resistance per unit length is `lambda`, then find the following.

`{:i)` The direction of induction current
`{:ii)` The current in the loop
`{:iii)` Potential difference between `P` and `Q`

Explanation

`{:i)` Direction of current is closewise.
`{:ii) |e| = (d phi)/(dt) = (d)/(dt)(BA) =a^(2)(alpha B_(0).t) = a^(2)alpha B`
Current : `i = (e)/(R ) = (a^(2)alpha B_(0))/(4 alpha lambda)(because R = 4 alpha lambda) = (a alpha B_(0))/(4 lambda)`
`{:iii) v_(p) + (e)/(4)-i. a lambda = V_(Q)`, where `'e'` is the total emf induced
or `V_(P)-V_(Q) = i a lambda - (e)/(4)`
or `V_(P)-V_(Q) = (e)/(4 a lambda).a lambda-(e)/(4), V_(P)-V_(Q) = (e)/(4)-(e)/(4) = 0`