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III. Magnetism
1. Electromagnetism in the laboratory and around us
2. Electromagnetism is simple. (If you know what it is!)
•It is about:
q - electric charges
(magnetic charges do not exist)
F - electromagnetic forces
E - electric fields
B - magnetic fields
•The two most important questions:
(1) How to find the force, F on the electric charge, q
exerted by the field E and/or B?
(2) How fields E and/or B can be created?
•Answers:
(1)


 
F  qE  qv  B

v - velocity of the charge q
(2) Four Maxwell's equations
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3. Electromagnetic forces
a) Electric forces (review):


FE  qE
Example1: (field created by one charge)
Ek
Q
r
F k
2
qQ
r2
Example2: (field created by a group of charges)
Two positive particles of equal charge, Q, are located at the corners of an equilateral
triangle of side d.
What is the electric field at the third corner of the triangle?
What is the electric force on a positive charge, q placed at the third corner?
E1  E2  k
Q
d2
  
E  E1  E2 ;
Q
E  3k 2 ;
d
E  2 E1 cos 30  2k
qQ
F  qE  3k 2 ;
d
Q 3
d2 2
 
F || E
2
b) Magnetic forces:

 
FB  qv  B

FB 
q

v


B


 - angle between v and B
Magnitude:
Direction:
FB  q vBsin 
   
F v , F B
+
right-hand rule
 
If v || B, i.e,   0  FB  0
 
If v  B, i.e,   90   maximum value of FB
Unit of B (the tesla):
[T]= [N/A·m]
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c) Electric and magnetic forces:
Electric
Magnetic
The first and the simplest – Coulomb’s law
F k
qQ
Ek
Q
r
q
2
always
FE  q E

v


B

 
FB  qv  B


FE  qE
r2
only for v=0

FB 
FB  q vBsin 


 
F  qE  qv  B
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4. Motion of charged particles in electromagnetic field
 
a  F /m
a) Electric field (review)
E
+
-
b) Magnetic field


FE  qE
  
a || FE || E
 

a || FB  v ;

  
a , FE , E - parallel to each other

 
a || FB  B;

FB  q vB sin 


 - angle between v and B
• Work done by magnetic force is zero!
(because magnetic force is perpendicular to displacement)
r
t cos   Fvt cos   0
t



  90 - angle between v and F
W  Fr cos   F
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c) Uniform magnetic field
1) If a charged particle is moving perpendicular to a uniform magnetic field,
its path will be a circle.
(Magnetic force is perpendicular to the velocity, so it is a centripetal force.)
 
v  B    90  sin   1
v
F

B
R
2 v
2R

T
; 
T
R
v
FB  q vB
v2
FB  m
R
v2
q vB  m
R
mv
R
qB
qB

m
2) If a charged particle in a uniform magnetic field is moving not
perpendicular to the field (ϴ ≠90°), than its path will be helical.
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

F  0  v  const
d) Velocity selector
 
qv  B
q0

qE
+


 
F  qE  qv  B  0 
E
v
B

B

v
-
e) Thompson’s experiment (discovery of electron)

v
-

v
+
+

B
-
V
1
2
mv  eV
2
2
e v

m 2V
E
v
B
e
E2

m 2VB 2
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Example: An electric field of 1.60 kV/m and magnetic field of 0.400 T act on a moving
electron to produce no net force. Calculate the minimum speed of the electron.


 
F  qE  qv  B  0 
E  1.60kV / m
B  0.400T
v min
v?
E 1.60  10 3 V / m
 
 4.0  10 3 m / s
B
0.400T
Example: Positive charge q is moving with velocity v in the magnetic field B as
shown below (with vectors v and B in the surface of the page). What is the direction
of the magnetic force?

FB 
q

v


B
Example: A positively charged particle is moving in a magnetic field. What is
the angle between the velocity of the charge and the direction of the magnetic
field if the magnetic force on the particle is zero?
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