Magnetic field – Ampere’s law – Lorentz force, cyclotron frequency, – Hall effect – Dipole moment, circulation electron, spin
NNSE 508 EM Lecture #2
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Magnetostatics: Ampere’s Law of Force •
•
•
Ampere’s law of force is the “law of action” between current carrying circuits through magnetic field. Experimental facts: – The magnitude of the force is inversely proportional to the distance squared. – The magnitude of the force is proportional to the product of the currents carried by the two wires.
•
The force acting on a current element I2 dl2 by a current element I1 dl1 is given by
F12
0
4
I 2 l2
I1 l1 aR12 R122
Experimental facts: – Two parallel wires carrying current in the same direction attract.
F21 F12 I1
I2
– Two parallel wires carrying current in the opposite directions repel. F12 F21
I1
I2
– A short current oriented perpendicular to another current experiences no force.
Permeability of free space unit vector in direction -7 2 10 {FW/m} ={N/A } 0=4 of I2 from I1
F12 = 0 I1
I2
NNSE 508 EM Lecture #2
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Magnetic field F12 •
0
I 2 l2
I1 l1 aR12 R122
4
The total force acting on a circuit C2 having a current I2 by a circuit C1 having current I1 is given by
II 4
F
F •
•
C2
dF
B
Idl
R122
I1dl1 aR12
0
4
C1
I1dl1 aR12
0
4
dl1 aR12
C2 C1
I 2 dl2
Similar to Coulomb’s law, magnetic field can be introduced
An infinitely small current element Idl immersed in a region of magnetic flux density B, experiences a force dF
dl2
0 1 2
C1
2 12
R
R122
T
B NNSE 508 EM Lecture #2
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The Biot-Savart Law •
The contribution to the B-field at a point P from a differential current element Idl is given by
dB(r ) •
R 0 I dl 3 4 R
P
R
Idl
r
r'
The total magnetic flux at the point P due to the entire circuit C is given by
0
B(r ) C
4
I dl
R R
3
NNSE 508 EM Lecture #2
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Lorentz Force
dF •
Idl B Idl
A moving point charge placed in both electric and magnetic field experiences a force given by
F
Q
Fe
v
Fm
Qv
q e v B
B
The force experienced by the point charge is in the direction into the paper. NNSE 508 EM Lecture #2
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Cyclotron Frequency • •
Electron circulating in a uniform magnetic field Magnetic field causes a centripetal force
F •
q v B
r
Angular frequency does not depend on electron velocity
v r
c
•
qvB
mv 2 r
qB m
c
Cyclotron resonance (CR) used for effective mass measurements, in particle accelerators (cyclotrons, synchrotrons).
2
mv qB qB 2 m Si
Cyclotron frequency Large mean free path of carriers (long scattering times) is needed for CR measurements :
c
1 NNSE 508 EM Lecture #2
Hall effect: carrier charge and concentration determination
F
Lorentz force on a moving particle:
Steady state: balance of forces in y direction:
7
ev B
eE y
evx Bz
Consider one type of carriers (e.g. n >> p)
Jx
Current density in x-direction:
envx
Hall coefficient (Hall and drift mobilities considered equal):
RH
Ey J x Bz
1 ne 1 pe
for for
n type p type
NNSE 508 EM Lecture #2
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Ampere’s Circuital Law in Integral Form •
•
•
Ampere’s Circuital Law in integral form states that the circulation of the magnetic field is proportional to the total current through the surface bounding the path over which the circulation is computed. Just as Gauss’s law follows from Coulomb’s law, so Ampere’s circuital law follows from Ampere’s force law.
B dl
I
0 encl
C
dl dS S
Just as Gauss’s law can be used to derive the electrostatic field from symmetric charge distributions, so Ampere’s law can be used to derive the magnetostatic field from symmetric current distributions. By convention, dS is taken to be in the direction defined by the right-hand rule applied to dl.
I encl
J ds S
Since volume current density is the most general, we can write Iencl in this way. NNSE 508 EM Lecture #2
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Applications of Ampere’s Law •
•
•
Ampere’s law in integral form is an integral equation for the unknown magnetic flux density resulting from a given current distribution. For example, magnetic field around a straight wire with a current I at a distance r:
BL B
•
Field outside is small If core with relative permittivity inserted
B
B dl C
B 2 r
Another example: field inside a solenoid is determined by a number of turns per length n and current I (no core):
known
I
0 encl
unknown
0
I
B
I 2 r 0
NI 0 nI 0
is
0
nI NNSE 508 EM Lecture #2
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Applications of Ampere’s Law: Coaxial cable •
•
Coaxial cable: thin internal wire carriers current I and coaxial metal cylinder carries –I Field is confined inside the outer cylinder
B dl
I
0 encl
C
NNSE 508 EM Lecture #2
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Magnetic dipole moment • •
•
A magnetic dipole comprises a small current carrying loop. The point charge (charge monopole) is the simplest source of electrostatic field. The magnetic dipole is the simplest source of magnetostatic field. The magnetic dipole is analogous to the electric dipole. Torque on a current loop in a uniform filed:
dF
Idl
B
B • •
The magnetic dipole moment can be defined as current times the area of the loop Similar to an electric dipole: B B
0
4
3
r r r5
r3
U
from
B to
B
IAn
Am 2
Direction of the dipole moment is determined by the direction of current using the right-hand rule . NNSE 508 EM Lecture #2
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Circulating electron and spin • Electron circulating with uniform angular speed produces current:
I
IA
• Magnetic moment:
• An electron also has an intrinsic magnetic dipole moment associated with its spin S:
q
2
1 q R2 2
q L 2m
• Can be expressed through angular momentum L m R2
• A magnetic dipole precesses about the direction of constant magnetic field
charge period
dL dt B
For negative charge m and L are in opposite directions
d dt
q gs S 2m
q 2m
B
Gyromagnetic ratios
with gs = 2.0023
NNSE 508 EM Lecture #2
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Test questions • Consider Hall effect in two samples with different carrier concentration. Which sample will show higher magnitude of Hall voltage?
