Download Introduction Last year we studied the electric and the magnetic field

Introduction
Last year we studied the electric and the magnetic field, now we will study the electromagnetic
field.
I remind you that in each point of an electric field, we considered a vector E which is defined by
the ratio between the force F applied on an electrical charge q and the electrical charge itself:
E
F
. In the same way, in each point of a magnetic field, we considered a vector B accordingly
q
to the Lorentz’s formula F  qv  B , where F is the force applied on an electrical charge q that
moves with velocity v .
Several experiments showed the existence of an electromagnetic field T defined in each point of
the space-time by 16 components, some of which are the Cartesian components of vector E and
vector B ; that is:

 0

 E x
T  c
 Ey

 c
 E z
 c
Ex
c
Ey
0
 Bz
Bz
0
 By
Bx
c
Ez 

c 
By 
 where c is the speed of the light in vacuum.

 Bx 

0 

T is called electromagnetic stress tensor. This is one of the most important results of Einstein’s
special relativity.
There are some relationships between the components of the tensor T that we need to study .
 E  d  
1) Faraday-Lenz’s law (also called induction law):

dt


 I   d S ( E ) 
B

d



0
0



dt


2) Ampere-Maxwell’s law:
What’s  E  d ? It’s called circulation of E :

d S ( B)
 E  d : lim

n 
n
E
i
 d i where d i is the
i 1
infinitesimal side of polygonal that approximate  . It is also the emf (electromotive force).
This is a way to say the 1st law: “The circulation of E along  is equal to the opposite of the
derivative of the magnetic flux with respect to time”.
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This is a way to say the 2nd law: “The circulation of B along  is equal to 0 multiplied by the sum
of the intensity of the conductions current and the product of 0 with the derivative of the electric
flux with respect to time”.
REM:  S ( B) : lim
n 
n
B
i
 dS i where dS i is the infinitesimal face of polyhedron that
i 1
approximates an open surface having boundary .
To demonstrate the validity of 1) above we have to change the magnetic flux. This is made in
several ways:
1.1 By a relative motion between a magnet and an electric circuit:
 A linear magnet going into a coil attached to an analog voltmeter;
 A spherical magnet going into a coil connected to a led;
 A linear magnet (hung on a spring) going up and down into a coil attached to the GLX
digital voltmeter.
1.2 By Variation of the magnetic intensity:
 Switching on and off an electromagnetic transformer;
 Throwing an aluminum ring (electromagnetic gun).
The meaning of “” in Lenz’s formula (to be grammatically corrected by someone)
If the surface vector point in one direction, as it’s shown here:
S
Then the positive circulation on the boundary is anti-clockwise:
V
Now we consider a linear magnet coming towards a conductive ring:
S
N
The magnetic flux is increasing as magnet moves towards the ring. So the derivative of the flux
with respect to time is positive, so the circulation is clockwise. Then the side of the ring in front of
the incoming North is a North pole too: there is a repulsive force that damps the motion of the
magnet.
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About mathematical operations for physics
“=” equals , is equal to
“” is not equal to
“<” is less than
“”is less than or equal to
“>” is greater than
“” is greater than or equal to
Addition:
“” is approximately equal to
“” infinity
“+” (plus) a + b = c , a plus b is equal to c , the sum of a and b is c
Subtraction:
“” (minus) a  b = c , a minus b equals c , the difference between of a and b is c
Multiplication:
“” (point) a  b = c , a times b equals c , a multiplied by b equals c, the product of a and b is c
Division:
“:” (colon) a : b = c , a divided by b is equal to equals c, the ratio/quotient of a and b is c
a/b ( a slash b) , (this fraction can be said as) , a divided by b , a over b
Note that “per” is similar in meaning to “divided by” and that “per” is only used for a quantity of
“one”, so you can say “per litre” but not “per two litres”. Also, “a” sometimes means “divided by”,
as in “When I tanked up, I paid € 10 for seven litres, so the fuel was € 1.43 a litre”
The vector product and the scalar product are the two ways of multiplying vectors which see the
most application in physics and astronomy. The magnitude of the vector product of two vectors can
be constructed by taking the product of the magnitudes of the vectors times the sine of the angle
(<180 degrees) between them. The magnitude of the vector product can be expressed in the form:
and the direction is given by the right-hand rule. If the vectors are expressed in terms of unit vectors
i, j, and k in the x, y, and z directions, then the vector product can be expressed in the rather
cumbersome form
which may be stated somewhat more compactly in the form of a determinant.
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The direction of the vector product can be visualized with the right-hand rule. If you curl the fingers
of your right hand so that they follow a rotation from vector A to vector B, then the thumb will
point in the direction of the vector product.
The vector product of A and B is always perpendicular to both A and B. Another way of stating that
is to say that the vector product is perpendicular to the plane formed by vectors A and B. This righthand rule direction is produced mathematically by the vector product expression.
The scalar product and the vector product are the two ways of multiplying vectors which see the
most application in physics and astronomy. The scalar product of two vectors can be constructed by
taking the component of one vector in the direction of the other and multiplying it times the
magnitude of the other vector. This can be expressed in the form:
If the vectors are expressed in terms of unit vectors i, j, and k along the x, y, and z directions, the
scalar product can also be expressed in the form:
The scalar product is also called the "inner product" or the "dot product" in some mathematics texts.
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