606
13 Electromagnetism I
Using (4) in (3)
d
2
=
2mV
eB 2
(5)
Thus the condition that the electrons are able to arrive at the positive plate is
d
2
<
2mV
eB 2
13.3.2 Magnetic Induction
13.21 B =
μ 0 i
2πr
∴
i =
2π Br
μ 0
=
2π × 10 −6 × 0.4
4π × 10 −7
= 2 A
13.22 (a) Consider a typical current element dx. The magnitude of the contribution dB of this element to the magnetic field at P is found from Biot–
Savart law and is given by (Fig. 13.22)
Fig. 13.22 Magnetic
induction due to a
current-carrying wire of finite
length
dB =
μ 0 i dx sin θ
4π R 2
(1)
Since the direction of the contribution dB at point for all such elements
is identical, i.e. at right angles to the plane of paper, the resultant field is
obtained by integrating dB from A to D in (1). Writing sin θ =
r
R
B =
dB =
μ 0 ir
4π
⎡
⎣
b
−a
dx
(x 2 + r 2 ) 3/2
⎤
⎦ =
μ 0 ir
4πr 2
x
(x 2 + r 2 ) 1/2
b
|
−a
13 Electromagnetism I
Using (4) in (3)
d
2
=
2mV
eB 2
(5)
Thus the condition that the electrons are able to arrive at the positive plate is
d
2
<
2mV
eB 2
13.3.2 Magnetic Induction
13.21 B =
μ 0 i
2πr
∴
i =
2π Br
μ 0
=
2π × 10 −6 × 0.4
4π × 10 −7
= 2 A
13.22 (a) Consider a typical current element dx. The magnitude of the contribution dB of this element to the magnetic field at P is found from Biot–
Savart law and is given by (Fig. 13.22)
Fig. 13.22 Magnetic
induction due to a
current-carrying wire of finite
length
dB =
μ 0 i dx sin θ
4π R 2
(1)
Since the direction of the contribution dB at point for all such elements
is identical, i.e. at right angles to the plane of paper, the resultant field is
obtained by integrating dB from A to D in (1). Writing sin θ =
r
R
B =
dB =
μ 0 ir
4π
⎡
⎣
b
−a
dx
(x 2 + r 2 ) 3/2
⎤
⎦ =
μ 0 ir
4πr 2
x
(x 2 + r 2 ) 1/2
b
|
−a