A rod PQ of length 1 is moved in uniform magnetic field \vec{B} as shown. What will be the emf induced in it?
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Answer:
The maximum emf induced in the coil will be
Explanation:
3
2
T
π
2
NB
0
(a
2
+ab+b
2
)
The number of turms in he spiral coil per unit radial width n=
b−a
N
The emf induced in an 'almost circular' part of spiral at radial distance r is
dV=
dt
dϕ
=
dt
d(nB.πr
2
)
=nB
0
T
2π
cos(
T
2π
t)πr
2
Since these are almost circular, but actually end to end connected, each of these rings at radial distances varying from a to b would add up to given total emf across the loop.
Hence V=∫
a
b
nB
0
T
2π
cos(
T
2π
t)πr
2
dr
=
3
2
T
π
2
NB
0
(a
2
+ab+b
2
)
cos(
T
2π
t) [Since (b
3
−a
3
)=(b−a)(a
2
+ab+b
2
)]
Thus the maximum value of the emf is
=
3
2
T
π
2
NB
0
(a
2
+ab+b
2
)
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