8.1.The Defenition of Magnetic Induction
Fe =
In which the assumption that the two charge are at rest is implicit.
If the charges are moving with constant velocities v and v1, respectively, an additional magetic force Fm is exerted on q by q1
Fm =
The number plays the same role here as 1/ played in electrostatics-that is, it is the constant required to make an experimental law compatible with a set of units. In mks units, by definition,
= 10-7 N.S2/C2
Fm = qv x B
Where the magnetic induction B is
If both an electric field and a magnetic field are present, the total force on a moving charge is Fe + Fm
F = q(E + V x B)
This force is known as the Lorentz Force
8.2.Forces on Current-Carrying Conductors
dF = NA │dI│qv x B
where A is the cross-sectional area of the conductor and q is the charge per charge carrier.
This expression can be used to obtained
as an alternative expression for the magnetic dipole moment.
8.3.The Law of Biot and Savart
In 1820, just a view weeks after Oersted announced his discovery that currents produce magnetic effect, ampere presented the results of a series of experiments that may be generalized and expressed in modren mathematical language as
F2 =
8.4.Elementary Application of The Biot and Savart Law
B(z) =
The first two terms integrate to zero, leaving
B(z) =
Which is, of course, enirely along the z-axis.
8.5.Ampere’s Circuital Law
Whit │B│as given above,
Ampere’s circuital law is an many ways parallel to Gauss’s law in electrostatics. That is, it can be used to obtain the magnetic field due to a certain current distribution of high symetry without having to evaluate the complicated integrals that appear in the Biot law.
8.6.The Magnetic Vector Potential
The calculation of electric fields was much simplified by the introduction of the electrostatic potential. The possibility of making this simplification resulted from the vanishing of the curl of the electric field. The curl of the magnetic induction does not vanish, however, its divergence does.
Since the divergence of any curl is zero, it is reasonable to assume that the magnetic induction may be written
The vector field A is called the magnetic vector potential.
The only requirment placed on A is that
x B = x x A = - J
Using this density
x x A = . A - 2A
And specifying that . A = 0 yields
2A = - J
Integrating each rectangular component and using the solution for Poisson’s equation as guide leads to
A(r2) =
The integrals involved in this expression are much easier to evaluate than those involved in the Biot law. However, they are also more complicated than those used to obtain the electrostatic potential.
8.7.The Magnetic Field of A Distant Circuit
A(r2) =
For circuits whose dimensionsare small compared with r2, the denominator can be approximated.
-1 =
And expand power of r1/r2 to get
8.8.The Magnetic Scalar Potential
Therefore, the magnetic induction in such regions can be written as the gradient of a scalar potential:
is called the magnetic scalar potential
8.9.Magnetic Flux
The quantity
Is known as the magnetic flux and is measured in webers (Wb). It is analogous to the electric flux discussed earlier, but it is of much greater importance.
The flux through a closed surface is zero, as can be seen by computing
It follows that the flux through a circuit is independent of the particular surfaced used to compute the flux
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