now let us understand, magnetic field induction
and its correlation with magnetic flux. magnetic
flux is a similar physical quantity which
we already discussed in electro statics that
is electric flux. now about magnetic flux,
its calculation is also the similar phenomena,
like we can say in a uniform, magnetic field,
through a given area, magnetic flux, can be
calculated as, in this situation we can see
if this is a uniform magnetic field of induction
b, in which all magnetic lines are parallel
and iquidistant. say we consider a normal
area placed in this region, and the total
area is s for this section, and we wish to
find out the total magnetic flux which is
passing through this normal area then we can
directly write, here magnetic flux phi is
equal to b s, because here also we define
the magnetic induction as the flux density,
so b we can write as phi by s if phi is the
flux passing through it. and if the area is
not normal, it is inclined at some angle then
we’ll consider its component like, say if
this is an area which is inclined from the
normal direction to, magnetic field by an
angle theta. and say this area is s, then
we can say, we resolve it in 2 perpendicular
components, like if this area is s this area
will be s sine-theta and, the area which is
normal to the direction of magnetic field,
the component of this area which is normal
to magnetic field is, s coz-theta. so in this
situation we can say, whatever flux is passing
through this area s coz-theta will be passing
through the total area, and its area vector
we can say from which the flux is coming out
is normal to the surface, this is s vector
and this is direction of magnetic induction
vector. the angle between the 2 is also theta.
so here we can write magnetic flux through
this area phi is equal to b s coz-theta because,
the flux density or magnetic induction is
the flux per unit normal area. so in this
situation this flux we can write as, b dot
s, which is almost the similar way, which
we’ve already applied in calculation of
electric flux, that if electric field strength
is e through a given area, the electric flux
is e dot s in uniform electric field. similar
to that here if magnetic induction is non
uniform, we can calculate magnetic flux, lets
see on the next sheet.
if we talk in, non uniform magnetic induction,
we can say in non uniform field, if we talk
about the magnetic flux, this is the magnetic
induction b vector which is not uniform in
this region it is varying, flux density is
different at different points. and here if
we consider a given area, and say we consider
a small elemental area d s over here, then
the direction of d s vector will be normal
to this, this’ll be d s vector. at this
point the direction of magnetic induction
will be tangential to any magnetic line passing
through d s. if this angle is theta, here
we can write let d phi, be the magnetic flux,
through elemental area d s. this implies we
can write d phi is equal to b dot d s because,
for the elemental area d s we can consider
b is not varying in such a small region, or
it can be written as b d s coz-theta. and
we can calculate the total flux, through surface,
by integrating this value like here phi will
be integration of d phi that’ll be integration
of b dot d s, or it is integration of b d
s coz-theta. the limit will be for the surface
m if we name the surface as m. here we should
always keep in mind that in non uniform field,
flux can be calculated by using the expression
integration of b dot d s. and if the field
is uniform that can be directly calculated
by integration of b dot s.
