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The Magnetic Field
Read: Chapter 18
The Basics
~ are generated by moving charges. When speaking of moving charges, we use the word current. The
Magnetic fields B
electron current i is the number of electrons per second that enter a section of conductor. Current is the (usually fictional)
flow of positive charge, and is opposite the direction of electron current.
Magnetic field lines form closed loops, unlike electric field lines which originate and terminate on charges. An experimentally
determined relation called the Biot-Savart Law descibes the magnetic field of a moving point charge:
~ = µ0 q~v × r̂ ,
B
4π r2
where µ0 is known as the permeability of free space (vacuum).
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The Magnetic Field
The Cross Product
~ =
B
µ0 q~
v ×r̂
4π r 2
The Biot-Savart Law involves a cross product (or vector product). The cross
product is perpendicular to the plane spanned by its component vectors. Given
~ and B,
~ the magnitude of A
~×B
~ is
vectors A
~ × B|
~ = |A||B| sin θ ,
|A
~ and B.
~
where θ is the angle between A
r̂
θ
q
+
~v
More generally, the cross product is given by the determinant
x̂
ŷ
ẑ ~×B
~ = Ax Ay Az A
Bx By Bz = (Ay Bz − Az By )x̂ + (Az Bx − Ax Bz )ŷ + (Ax By − Ay Bx )ẑ .
Review Exercise: Use the determinant to calculate the z component of angular momentum.
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The Magnetic Field
The Cross Product
The direction of the cross product may be determined using a right hand rule (RHR). The procedure goes like this for the
Biot-Savart law:
1. point the fingers of your right hand in the direction of ~v .
2. curl your fingers in the direction of r̂.
~
3. your thumb now points in the direction of B.
Since cross products are inherently three-dimensional while paper (and screens) are not, we use the following conventions:
• vectors pointing out of the page are denoted .
• vectors pointing into the page are denoted ⊗.
~ × B.
~
Exercise: Use the RHR to determine the direction of A
y
y
y
y
~
B
~
B
z
~
A
x
~
A
z
~
B
y
~
A
~
A
x
z
x
~
B
z
3
~
A
x
x
z
~
B
The Magnetic Field
Current
A current is generated by a non-zero electric field inside a conductor — in other words, a current-carrying wire is not in
~ int 6= 0). Typically, current carrying conductors are neutral ; charge is moving, but there is no excess charge.
equilibrium (E
~ net , where u is the mobility of the
Recall the concept of drift velocity: v̄ = uE
~
material. Under influence of Eext , mobile charges quickly reach v̄ and a current
is established.
v̄
~ ext
E
In a neutral current carrying wire with mobile electron density n, cross-sectional
~ we define the electron current i — the flow of
area A, and electric field E,
electrons per unit time — by
~ .
i = nAv̄ = nAu|E|
A
v̄∆t
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The Magnetic Field
Current
~ usually referred to simply as current. Even
For problem solving, physicists and engineers use conventional current I,
though no positive charges are flowing in a circuit, we solve problems as though there are. In terms of physical quantities,
~ C/s .
I~ = |q|nAuE
~
∆B
The Biot-Savart law can be applied to currents, but we must use an infinitesimal
~ The magnetic field
form. Consider a short, segement of wire ∆l with current I.
~
∆B at some location ~r with respect to the wire is given by
~ =
∆B
r̂
θ
∆~l
I~
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µ0 ∆lI~ × r̂
4π r2
The Magnetic Field
Long Thin Wire
~ of a current distribution is just like that for a chage
The procedure calculating B
distribution, only the charge element ∆Q is replaced by the wire segment ∆l.
I~
~
1. Divide into ∆l and draw ∆B.
L/2
~
2. Write an expression for ∆B.
∆l
y
~r
x
~
∆B
p
3. Sum (integrate) contributions of ∆l (dl).
4. Check the result.
z
−L/2
The end result of the calculation yields
~ wire | =
|B
≈
LI
µ0
p
4π r r2 + (L/2)2
µ0 2I
if L r .
4π r
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The Magnetic Field
Circular Loop
I~
To calculate B along the axis of a circular wire loop of radius R, we follow
the same procedure as for the thin wire. Note the calculations for current
distributions are very similar to previous calculations for E.
x
~
1. Divide into ∆l and draw ∆B.
∆l
~
2. Write an expression for ∆B.
∆θ
~r
θ
p
y
~
∆B
3. Sum (integrate) contributions of ∆l (dl).
4. Check the result.
z
rz
The result is
~ loop | =
|B
2πR2 I
µ0
.
4π (z 2 + R2 )3/2
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The Magnetic Field
Current Ring
The current ring produces a dipole field pattern. This suggests we define a
~
magnetic dipole moment µ
~ . Looking at the equation for |B|,
~ loop | =
|B
µ0
2πR2 I
.
4π (z 2 + R2 )3/2
we see πR2 I = IA in the numerator.
The magnetic dipole moment is defined as
~.
µ
~ ≡ IA
~ is found using another RHR: curl the fingers of your right hand in the direction of conventional current
The direction of A
~
around the loop — your extended thumb points in the direction of A.
8
The Magnetic Field
Bar Magnets
A bar magnet also produces a dipole field. At points far from the magnet, the
field may be approximated along the axis of the magnet using the field of a
loop,
~ bar | ≈ µ0 2µ ,
|B
4π r3
where µ is determined experimentally.
There are no magnetic monopoles. In other words, there are no “magnetic charges” for field lines to start from or
terminate on. Consequently, magnetic field lines always form closed loops.
~ earth | =
Example: Antarctic Circuit, Ch. 18, pg. 728. R = 5 cm, I = 5 A, µbar = 1.2 Am2 , |B
−5
6 × 10 T.
∆l
r̂
θ
A
R
I~
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The Magnetic Field
Atomic Level
A single atom can be a magnetic dipole. A simple model of H has an electron in a
circular orbit about the proton, so has dipole moment
−
|~ve |
1
|~
µe | = e
πR2 = eR|~ve | .
2πR
2
+
R
~ e , this becomes
In terms of orbital angular momentum L
|~
µe | =
1 e ~
|Le | .
2 me
Orbital angular momentum of the electron is quantized in units of ~: L = N ~. Assuming N = 1 and ignoring spin angular
momentum, our expression becomes
e~
|~
µe | =
≈ 1 × 10−23 Am2 /atom .
2me
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