Chapter 29: Creating Magnetic Fields
PHY2049: Chapter 29
1
Creating Magnetic Fields
ÎSources
of magnetic fields
‹ Spin
of elementary particles (mostly electrons)
‹ Atomic orbits (L > 0 only)
‹ Moving charges (electric current)
ÎCurrents
generate the most intense magnetic fields
‹ Discovered
ÎThree
by Oersted in 1819 (deflection of compass needle)
examples studied here
‹ Long
wire
‹ Wire loop
‹ Solenoid
PHY2049: Chapter 29
2
B Field Around Very Long Wire
ÎField
around wire is circular, intensity falls with distance
‹ Direction
given by RHR (compass follows field lines)
µ 0i
B=
2π r
µ0 = 4π ×10−7
Right Hand Rule #2
PHY2049: Chapter 29
3
Long Wire B Field Example
ÎI
= 500 A toward observer. Find B vs r
‹ RHR
⇒ field is counterclockwise
µ i ( 4π ×10 ) 500 0.0001
B=
=
=
−7
0
2π r
‹r
‹r
‹r
‹r
‹r
‹r
‹r
=
=
=
=
=
=
=
0.001 m
0.005 m
0.01 m
0.05 m
0.10 m
0.50 m
1.0 m
2π r
B
B
B
B
B
B
B
=
=
=
=
=
=
=
r
0.10 T
0.02 T
0.010 T
0.002 T
0.001 T
0.0002 T
0.0001 T
=
=
=
=
=
=
=
1000 G
200 G
100 G
20 G
10 G
2G
1G
PHY2049: Chapter 29
4
Charged Particle Moving Near Wire
ÎWire
carries current of 400 A upwards
moving at v = 5 × 106 m/s downwards, 4 mm from wire
‹ Find magnitude and direction of force on proton
‹ Proton
ÎSolution
of force is to left, away from wire
‹ Magnitude of force at r = 4 mm
‹ Direction
 µ0 I 
F = evB = ev 

2
π
r


(
F = 1.6 × 10−19
)(
−7

2
10
×
× 400 
6
5 × 10 


0.004


)
F = 1.6 × 10−14 N
PHY2049: Chapter 29
v
I
5
Ampere’s Law
ÎTake
arbitrary path around set of currents
‹ Let ienc be total enclosed current (+ up, − down)
‹ Let B be magnetic field, and ds be differential length
∫ B ⋅ ds = µ0ienc
ÎOnly
along path
Not included
in ienc
currents inside path contribute!
‹5
currents inside path (included)
‹ 1 outside path (not included)
PHY2049: Chapter 29
6
Ampere’s Law For Straight Wire
ÎLet’s
try this for long wire. Find B at distance at point P
‹ Use
circular path passing through P (radius r)
‹ From symmetry, B field must be circular
∫ B ⋅ ds = B ( 2π r ) = µ0i
B=
ÎAn
µ 0i
2π r
P
r
easy derivation
PHY2049: Chapter 29
7
Useful Application of Ampere’s Law
ÎFind
B vs r inside long wire, assuming uniform current
‹ Wire
radius R, total current i
‹ Find B at radius r = R/2
ÎKey
ienc
fact: enclosed current ∝ area
R
 π ( R / 2 )2  i
=
= i
2
 πR
 4


r
R
i

B  2π  = µ0
2
4

1 µ 0i
B=
2 2π R
µ0i
B=
2π R
PHY2049: Chapter 29
On surface
8
Force Between Two Parallel Currents
ÎForce
on I2 from I1
µ0 I1I 2
 µ0 I1 
F2 = I 2 B1L = I 2 
L=
L

2π r
 2π r 
‹ RHR
⇒ Force towards I1
ÎForce
on I1 from I2
µ0 I1I 2
 µ0 I 2 
F1 = I1B2 L = I1 
L=
L

2π r
 2π r 
‹ RHR
⇒ Force towards I2
ÎMagnetic
I2
I2
I1
forces attract two parallel currents
I1
PHY2049: Chapter 29
9
Force Between Two Anti-Parallel Currents
ÎForce
on I2 from I1
µ0 I1I 2
 µ0 I1 
F2 = I 2 B1L = I 2 
L=
L

2π r
 2π r 
‹ RHR
⇒ Force away from I1
ÎForce
on I1 from I2
µ0 I1I 2
 µ0 I 2 
F1 = I1B2 L = I1 
L=
L

2π r
 2π r 
‹ RHR
⇒ Force away from I2
ÎMagnetic
I2
I2
I1
forces repel two antiparallel currents
I1
PHY2049: Chapter 29
10
Parallel Currents (cont.)
ÎLook
B
at them edge on to see B fields more clearly
B
2
Antiparallel: repel
1
F
2
1
F
B
2
1
B
Parallel: attract
F
1
2
F
PHY2049: Chapter 29
11