MAGNETIC FLUX : `phi_B`
Magnetic flux `phi_B` through a surface placed inside a
Magnetic Filed `vecB` is defined as
`phi_B = int vecB. vec(dA) ............ (1)`
Where `vec(dA)` is an elemental area vector on the given
surface and `vecB` is the Magnetic Field vector at the given
point.
`text(IMPORTANT NOTE)`
`=>` If `vecB` is uniform (i.e. the magnitude and the direction of
the magnetic field is constant throughout space and also
constant in time) and the surface in consideration is a
plane 'lamina' (eg. a disc, rectangular surface etc),
equation (1) can be simplified to `phi_B = vecB*vecA ................ (2)`
Where `vecB` is the magnetic field and `vecA = Ahatn` is the area
vector for the laminar surface and ,tis a unit vector
'normal' to it
`=>` If `B` is uniform in space but varies with the time then we
use `(2)` but here `vecB` represents magnetic field at any
instant of time hence `phi_B` is a function of time
`=>` If `B` varies in space and with time then we use `(1)` but
`phi_B` becomes a function of time.
Physically, as in the case of the Electric Flux associated
with a surface inside an electric field, Magnetic Flux for a
surface, `phi_B` represents the total number (algebraic sum)
of lines of magnetic induction passing through a given
area.
`text(DIMENSIONS:)`
`[phi_B] =[B][A]= [F/(IL)][A]` , (Here `F` represents the force
acting on a current carrying conductor of length L placed
inside the magnetic field `B`)
`[phi_B] = [(MLT^-2)/(IL)][L^2] = [ML^2T^-2I^-1]`
`text(UNIT:)`
Since `[ ML^2T^-2I^-1]` represents the Sl unit for energy, the Sl
unit of magnetic flux is
(joule)/(ampere) = ((joule)u `xx` (sec))/(coulomb) [ as ampere = ((coulomb)/(sec))
ampere coulomb sec
=volt `xx` sec [ as joule/coulomb) =volt]
and is called Weber= (Wb) or T `xx m^2` (as Tesla `= (Wb)/ m^2` ).
The `CGS` unit of flux , Maxwell (Mx) is related to Weber
(Wb) through the relation
`1 Wb = 1 V xx S = 10^8` `Mx`
`text(CALCULATING MAGNETIC FLUX :)`
The magnetic flux associated with the given area in
a field `vecB` depends on the Magnetic field itself, `vecB`, the area
of the surface and also the angle `q` between the area
vector and the field, since for a given elemental area `vec(dA)`
and magnetic field `vecB`. flux is given by
`dphi_B = vecB*vec(dA) = BdA cos theta`
This is illustrated with the following example : Consider
an elemental area vector `vec(dA)` with the three different
orientations w.r.t the magnetic field `vecB` as shown
Thus magnetic flux is minimum i.e. `d phi_B = vecB*vec(dA) =0` for the
case when `theta = 90^o` (the last case). whereas magnetic flux
is maximum `dphi_B= vecB.vec(dA)` when `theta = 0` (second case)
Magnetic Filed `vecB` is defined as
`phi_B = int vecB. vec(dA) ............ (1)`
Where `vec(dA)` is an elemental area vector on the given
surface and `vecB` is the Magnetic Field vector at the given
point.
`text(IMPORTANT NOTE)`
`=>` If `vecB` is uniform (i.e. the magnitude and the direction of
the magnetic field is constant throughout space and also
constant in time) and the surface in consideration is a
plane 'lamina' (eg. a disc, rectangular surface etc),
equation (1) can be simplified to `phi_B = vecB*vecA ................ (2)`
Where `vecB` is the magnetic field and `vecA = Ahatn` is the area
vector for the laminar surface and ,tis a unit vector
'normal' to it
`=>` If `B` is uniform in space but varies with the time then we
use `(2)` but here `vecB` represents magnetic field at any
instant of time hence `phi_B` is a function of time
`=>` If `B` varies in space and with time then we use `(1)` but
`phi_B` becomes a function of time.
Physically, as in the case of the Electric Flux associated
with a surface inside an electric field, Magnetic Flux for a
surface, `phi_B` represents the total number (algebraic sum)
of lines of magnetic induction passing through a given
area.
`text(DIMENSIONS:)`
`[phi_B] =[B][A]= [F/(IL)][A]` , (Here `F` represents the force
acting on a current carrying conductor of length L placed
inside the magnetic field `B`)
`[phi_B] = [(MLT^-2)/(IL)][L^2] = [ML^2T^-2I^-1]`
`text(UNIT:)`
Since `[ ML^2T^-2I^-1]` represents the Sl unit for energy, the Sl
unit of magnetic flux is
(joule)/(ampere) = ((joule)u `xx` (sec))/(coulomb) [ as ampere = ((coulomb)/(sec))
ampere coulomb sec
=volt `xx` sec [ as joule/coulomb) =volt]
and is called Weber= (Wb) or T `xx m^2` (as Tesla `= (Wb)/ m^2` ).
The `CGS` unit of flux , Maxwell (Mx) is related to Weber
(Wb) through the relation
`1 Wb = 1 V xx S = 10^8` `Mx`
`text(CALCULATING MAGNETIC FLUX :)`
The magnetic flux associated with the given area in
a field `vecB` depends on the Magnetic field itself, `vecB`, the area
of the surface and also the angle `q` between the area
vector and the field, since for a given elemental area `vec(dA)`
and magnetic field `vecB`. flux is given by
`dphi_B = vecB*vec(dA) = BdA cos theta`
This is illustrated with the following example : Consider
an elemental area vector `vec(dA)` with the three different
orientations w.r.t the magnetic field `vecB` as shown
Thus magnetic flux is minimum i.e. `d phi_B = vecB*vec(dA) =0` for the
case when `theta = 90^o` (the last case). whereas magnetic flux
is maximum `dphi_B= vecB.vec(dA)` when `theta = 0` (second case)