628
13 Electromagnetism I
∴
dv = −
B 2 l 2
m R
ds
v =
dv = −
B 2 l 2 s
m R
+ C
where C = constant of integration.
When s = 0, v = u
∴
c = u
∴
s =
(u − v)m R
B 2 l 2
∴
s max =
u m R
B 2 l 2
where we have put v = 0.
13.81 ξ = −N
dφ
dt
= −N
d
dt
(B A) = −N πr
2 dB
dt
(Faraday’s law)
= −100 × π(0.1)
2
(0.1) = 0.314 V
13.82 (i) The flux on one side is equal and opposite to that on the other for 0 <
r < b − x, where r is the distance of any point from the long wire. Only
the flux through the portion b − x < r < x is not cancelled:
dφ = (adr )db =
μ 0 I adr
2π r
∴
φ =
μ 0 I a
2π
x
b−x
dr
r
=
μ 0 I a
2π
ln
x
b − x
(ii) φ → ∞ for x → b and φ = 0 for x = b/2
(iii) For I = 2t and x = b/4
φ =
μ 0 a
2π
ln
1
3
2t
ξ = −
dφ
dt
=
μ 0 a ln 3
π
13.83 Betatron is a machine to accelerate electrons to high energy. It consists of
an evacuated ‘doughnut’ in which the electrons are made to circulate under
the influence of changing magnetic field. If a magnetic flux φ changes in an
electromagnet, it accelerates the electrons and at the same time holds them
in an orbit of fixed radius. The average force acting on the particle during a
single rotation is the work (induced voltage multiplied by charge) divided by
the distance 2π R. Equating this to the time rate of change of momentum, by
Faraday’s law of induction
13 Electromagnetism I
∴
dv = −
B 2 l 2
m R
ds
v =
dv = −
B 2 l 2 s
m R
+ C
where C = constant of integration.
When s = 0, v = u
∴
c = u
∴
s =
(u − v)m R
B 2 l 2
∴
s max =
u m R
B 2 l 2
where we have put v = 0.
13.81 ξ = −N
dφ
dt
= −N
d
dt
(B A) = −N πr
2 dB
dt
(Faraday’s law)
= −100 × π(0.1)
2
(0.1) = 0.314 V
13.82 (i) The flux on one side is equal and opposite to that on the other for 0 <
r < b − x, where r is the distance of any point from the long wire. Only
the flux through the portion b − x < r < x is not cancelled:
dφ = (adr )db =
μ 0 I adr
2π r
∴
φ =
μ 0 I a
2π
x
b−x
dr
r
=
μ 0 I a
2π
ln
x
b − x
(ii) φ → ∞ for x → b and φ = 0 for x = b/2
(iii) For I = 2t and x = b/4
φ =
μ 0 a
2π
ln
1
3
2t
ξ = −
dφ
dt
=
μ 0 a ln 3
π
13.83 Betatron is a machine to accelerate electrons to high energy. It consists of
an evacuated ‘doughnut’ in which the electrons are made to circulate under
the influence of changing magnetic field. If a magnetic flux φ changes in an
electromagnet, it accelerates the electrons and at the same time holds them
in an orbit of fixed radius. The average force acting on the particle during a
single rotation is the work (induced voltage multiplied by charge) divided by
the distance 2π R. Equating this to the time rate of change of momentum, by
Faraday’s law of induction