UPSEE 2018 Solved Paper

Area vector of the given loop,
A=ab

^

k

Time dependent magnetic field,
B=c[(x‌cos‌ω‌t)

^

i

+(y‌sin‌ω‌t)

^

k

]
∴ Magnetic flux through the loop,
ϕn=A.B
=ab

^

k

[c{(x‌cos‌ω‌t)

^

i

+(y‌sin‌ω‌t)

^

k

}]
=abcy‌sin‌ω‌t ........(i)
∵0≤y≤b
∴yav=

0+b

2

=

b

2

Putting the value of y in Eq. (i), we get
∴ϕn=abc.

b

2

.sin‌ω‌t =

ab2

2

c‌sin‌ω‌t
∴ Induced emf in the loop,
ε=

d

dt

ϕn=

d

dt

.

ab2c

2

‌sin‌ω‌t
Hence, |ε|=|

ab2c

2

ω‌cos‌ω‌t|