Biot-Savart Law
ÎDeduced
from many experiments on B field produced by
currents, including B field around a very long wire
‹ Magnitude
dB =
µ0 i ds sin θ
4π
r2
Like Coulomb’s law
dB=0 ahead of ds and behind it.
Maximum on plane perp. to ds.
‹ Direction:
RHR #2
‹ Vector notation
r µ0 i dsr × rr
dB =
4π r 3
‹ Applications
1/r2 depedence
Reproduces formula for B around long, current-carrying wire
„ B by current loop (on axis)
„ In more complicated cases, numerically integrate to find B
„
PHY2049: Chapter 29
1
Law of Magnetism
ÎUnlike
the law of electrostatics, comes
in two parts
‹ Part
1
Effect of B field on moving charge
r
r
F = qv × B
‹ Part
2
Current produces B
equivalent
Biot-Savart Law
Ampere’s Law
Proof of equivalence not in the book
(Require vector calculus and relies on the
absence of magnetic monopoles)
PHY2049: Chapter 29
2
B Field on Axis of Circular Current Loop
ÎRadius
R and current i: find B field at
center of loop
µ 0i
B=
From B-S law by integration
2R
‹ Direction:
RHR #3 (see picture)
B=
µ0
i R2
2 R2 + z2
(
)
32
ÎIf
N turns close together
N µ 0i
B=
2R
ÎB field on axis, including center
µ0
i R2
B=
2 R2 + z2
(
‹ z=0:
)
32
From B-S law by integration
checks
µ0 i R 2
B=
‹ z>>R:
2 z3
Like E field around electric dipole!
PHY2049: Chapter 29
3
Current Loop Example
Îi
= 500 A, r = 5 cm, N=20
B=N
µ 0i
2r
=
( 20 ) ( 4π ×10−7 ) 500
2 × 0.05
= 1.26T
PHY2049: Chapter 29
4
Field at Center of Partial Loop
ÎDirection
of B?
ÎSuppose
partial loop covers angle φ
‹ Calculate
B field from proportion of full circle
µ 0i  φ 
B=


2 R  2π 
ÎUse
example where φ = π (half circles)
‹ Define
direction into page as positive
µ 0i  π
B=

2 R1  2π
 µ 0i  π 
−


2
2
π
R


2
µ 0i  1
1 
B=
 − 
4  R1 R2 
PHY2049: Chapter 29
5
Partial Loops (cont.)
ÎNote
on problems when you have to evaluate a B field at
a point from several partial loops
‹ Only
loop parts contribute, proportional to angle (previous slide)
‹ Straight sections aimed at point contribute exactly 0
‹ Be careful about signs, e.g. in (b) fields partially cancel, whereas
in (a) and (c) they add
PHY2049: Chapter 29
6
Solenoid and Toroid
ÎAnother
‹ Read
application of Ampere’s law
the book
PHY2049: Chapter 29
7
FAQ on Magnetism (2)
ÎAccording
to the law of magnetism, a current produces a
magnetic field, which exerts a force on a moving charge.
In the phenomenon of two bar magnets attracting each
other, I see no current in magnet 1 and no moving charge
in magnet 2, and vise versa. Doesn’t this example show
that the theory is incomplete?
‹ A:
magnet 1 comprises magnetic ions, which produce magnetic
field due to the orbital motion of electrons and the spins of
electrons. Magnet 2 also comprises magnetic ions, in which
electrons (negative charges) are orbiting around the nuclei and
electrons are also magnetic dipole moments. The force between
two magnets can be derived from the law, although the
calculation is lengthy and you first need to derive the formula for
the force exerted on a magnetic dipole by non-uniform B.
PHY2049: Chapter 29
8