A conducting square frame of side 'a' and a long straight wire carrying current I are located in the same plane as shown in the figure. The frame moves to the right with a constant velocity 'V'. The emf induced in the frame will be proportional to :

Options

A. `1/x^2`
B. `1/(2x-a)^2`
C. `1/"(2x+a)"`
D. `(1/((2x-a))-1/(2x+a))`

Answer

D

Explanation

Inside (1) magnitude of induced emf is `epsilon_1=B_1 Vl`
Inside (2) magnitude of induced emf is `epsilon_2=B_2 Vl`
Induced emf value in frame `epsilon=epsilon_1-epsilon_2` [Side (1) is near to wire and `B prop 1/r`]
`therefore epsilon=Vl (B_1-B_2)`
`therefore epsilon prop (B_1-B_2)`
but `B_1 prop 1/(x-a/2)` and `B_2 prop 1/(x+a/2)`
`therefore epsilon prop (1/(x-a/2)-1/(x+a/2))`
`therefore epsilon prop (2/(2x-a) -2/(2x+a))`
`therefore epsilon prop (1/(2x-a)-1/(2x+a))` ( `because` 2 constant)