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/2πr. The total magnetic flux ϕp through the loop ca n be obtained by summingover contributions from all differential are elements dA = ldr . So , ϕB = ∫dϕB = ∫
→
B
.d
→
A
=
µ0II
2Ï€
‌
s+w
∫
s
dr
r
=
µ0II
2Ï€
‌log(
s+w
s
) According to Faraday’s Law, the induced emf is ε =
dϕB
dt
=
d
dt
[
µ0II
2Ï€
loge(
s+w
s
)] =
µ0I
2Ï€
loge(
s+w
s
)
dI
dt
I = I (t) = a + bt ⇒
dI
dt
= b ⇒ ε =
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.