For calculating the induced e.m.f. we must first calculate the flux associated with the loop due to field of current carrying wire. The magnetic field due to a current – carrying wire at a distance r away is B=
µ0I
2Ï€r
. The total magnetic flux ϕB B through the loop can be obtained by summing over contributions from all differential are elements dA=ldr. So, ϕB=∫dϕB=∫
→
B
.d
→
A
=
µ0II
2Ï€
‌
s+w
∫
s
dr
r
=
µ0II
2Ï€
loge(
s+w
s
) According to Faraday’s Law, the induced emf is ξ=
dϕB
dt
=
d
dt
[
µ0II
2Ï€
loge(
s+w
s
)] =
µ0l
2Ï€
loge(
s+w
s
)
dI
dt
I=I(t)=a+bt⇒
dI
dt
=b ⇒ξ=
µ0bl
2Ï€
loge(
s+w
s
) The straight wire carrying a current I produces a magnetic flux into the page through the rectangular loop. By Lenz’s Law, the induced current in the loop must be flowing counter clockwise in order to produce a magnetic field out of the page to counteract the increase in inward flux.