In a uniform magnetic field, a `pi` shaped metal frame is located perpendicular to the plane of the conductor and varying with time at the rate `(dB//dt)=0.20T//s`. A conducting connector starts moving with an acceleration `a=30cm//s^(2)` along the parallel bars of the frame. The length of the connector is equal to l = 44 cm. Find the emf induced in the loop t = 1 s after the beginning of the motion, if at the moment t = 0, the loop area and the magnetic induction are equal to zero.

Explanation

We know, according to kinematic equation, distance travelled
`x=ut+(1)/(2)"at"^(2)`. Here, initial velocity, u = 0, acceleration,
a = b = `30cm//s^(2)`
So, in time t, distance travelled by conductor, `x=(1)/(2)bt^(2)`
Flux passing through loop, `phi=BS=Blx=(Blbt^(2))/(2)`
Here, B and t both are variable.
So, induced emf, `e=|(dphi)/(dt)|=(lb)/(2)[(dB)/(dt)t^(2)+B.2t]`
`=(0.4xx0.3)/(2)(0.2xx2^(2)+2B)`
`=0.06(0.8+2B)`
Given, `(dB)/(dt)=0.2T//srArr" "dB=0.2dt`
On integrating both sides, we get
`int_(0)^(B)dB=0.2int_(0)^(t)dtrArrB=0.2t=0.2xx1=0.2`
So, induced emf, `e=0.06(0.8+0.2)=0.06V=6xx10^(-2)V`