Magnetic Field Generated by Current in Straight Wire (1)
Consider a field point P that is a distance R from the axis of the wire.
µ0 Idx
µ0 Idx
sin
φ
=
cos θ
4π r2
4π r2
R
R
r2
dx
=
= 2 2 =
• x = R tan θ ⇒
dθ
cos2 θ
R /r
R
• dB =
µ0 I
µ0 I r2 dθ
cos
θ
=
cos θdθ
• dB =
4π r2 R
4π R
Z θ2
µ0 I
• B=
cos θdθ
4π R θ1
µ0 I
(sin θ2 − sin θ1 )
=
4π R
• Length of wire: L = R(tan θ2 − tan θ1 )
Wire of infinite length: θ1 = −90◦ , θ2 = 90◦ ⇒ B =
µ0 I
2πR
26/3/2008
[tsl216 – 3/16]
Magnetic Field Generated by Current in Straight Wire (2)
Consider a current I in a straight wire of infinite length.
• The magnetic field lines are concentric circles
in planes prependicular to the wire.
• The magnitude of the magnetic field at distance R
µ0 I
.
from the center of the wire is B =
2πR
• The magnetic field strength is
proportional to the current I and
inversely proportional to the distance R
from the center of the wire.
• The magnetic field vector is tangential
to the circular field lines and directed
according to the right-hand rule.
26/3/2008
[tsl217 – 4/16]
Magnetic Field Generated by Current in Straight Wire (3)
~ in the limit R → 0.
Consider the magnetic field B
• B=
µ0 I
(sin θ2 − sin θ1 )
4π R
• sin θ1 = √
a
a2 + R 2
= q
2a
1
1+
R2
a2
1 R2
≃1−
2 a2
1
1 R2
≃1−
2 4a2
• sin θ2 = √
4a2 + R2
µ0 I
• B≃
4π R
„
«
1 R2
1 R2
−1+
1−
2 4a2
2 a2
= q
1+
µ0 I 3R R→0
−→ 0
=
4π 8a2
R2
4a2
a
B
R
a
θ2
a
θ1
I
26/3/2008
[tsl380 – 5/16]