The quantity E is the electric-field vector, ds is a differential length about an
arbitrary closed path C, and ( is the magnetic flux through the surface defined by
the closed path C and is expressed in units of Wb. Faraday's Law states that the line
integral of the electric field about an arbitrary closed path C is equal to the negative
of the time-derivative of the magnetic flux through the surface defined by C. The
magnetic flux is defined as
S= jA.da (3-74)
The quantity A is the magnetic induction (or magnetic flux density) vector and
da is a differential area on an arbitrary open surface S, with da being normal to S. B
is expressed in units of Wb m 2 or T. Therefore, ( is equal to the surface integral of
the magnetic induction over the open surface S. If the open surface S is defined by the
closed path C, then Equation 3-73 can be re-written as
E .ds = Bda (3-75)
c dt s
Moreover, if the medium is stationary, then both C and S will be stationary, and
the expression can be further simplified further to
dA-
ds=- -- da (3-76)
Jc Jsdt
Hence, if the medium is stationary, then the line integral of the electric field
about the closed path C is equal to the negative of the surface integral (taken over the
surface defined by C) of the time-derivative of the magnetic induction. If the medium
is linear, homogeneous, and non-conducting (e.g. air), the magnetic induction, B, can
be expressed as the magnetic field intensity, H, times the permeability of the medium,
pu. If the medium is air, the value of u is approximately Po = 47n x 10 7 H m1 which is
the permeability of free space. Since B and H differ by only a constant in air, the term
"magnetic field" is used to refer to either H or B.