In the figure shown `'R'` is a fixed conducting fixed ring of negligible resistance and radius `'a'` `PQ` is a uniform rod of resistance `r`. It is hinged at the centre of the ring and rotated about this point in clockwise direction with a uniform angular velocity `w`. These is a uniform magnetic filled of strength `'B'` ponting inwards. `'r'` is a stationary resistance

Options

A. Current through `'r'` is zero.
B. Current through `'r'` is `(2B omega a^(2))/(5r)`.
C. Direction of current in external `'r'` is from centre to circumference.
D. Direction of current in external `'r'` is from circumference to centre. `k`

Answer

B, D

Explanation

Equivalent circuit :
Induced emf `e = (B omega r^(2))/(2) = ((B omega a^(2))/(2))`
(`Q` Radius a)
by nodal equation :
nodal
`4((X-e)/(r )) + ((X-0)/( r)) = 0`

`5X = 4e`
`Ϸ X = (4e)/(5) = (2B omega a^(2))/(5)` and `I = (X)/(r ) = (2B omega a^(2))/(5r)`
also direction of current in `'r'` will be towards negative terminal i.e., from rim to origin.
Alternating, by equivalent of cells (fig (ii)) :
`I = (e)/(r+(r )/(4)) = (4e)/(5r)`