Magnetic flux defined as “No of field lines intercepting perpendicularly to the plane (or) surface”.
Mathematically interprected by
\(phi =overrightarrow{B}centerdot overrightarrow{A}\) \(overrightarrow{B}\) – Magnetic induction \(overrightarrow{A}\) – Area vectorHere,
Area is a vector quantity, direction of the Area vector \((overrightarrow{A})\) is perpendicular to the plane (or) surface.
vFor open surface,
We can taken any one direction as Area vector \((overrightarrow{A})\) but we stick to that till the end of problem
vFor closed surface:
Area vector is outward normal.
Case I: [\(overrightarrow{B}\) is uniform] and [given surface is plane]
Look at the given plane, Area vector \((overrightarrow{A})\)
Of the plane is making angle (Ɵ) with direction of the magnetic induction \(left( overrightarrow{B} right)\) the magnetic flux for the plane can be
written as
φ = BAcosƟ
Here, we have resolved the \(overrightarrow{B}\) into its components BcosƟ and BsinƟ. Here BcosƟ is perpendicularly intercepting the given plane.
Case: II [\(overrightarrow{B}\) is uniform and given surface is arbitrary shape]
For any arbitrary surface, flux can be defined as by dividing the surface into small differential element area (dA) and flux linking the elemental area is (dφ)
dφ = B . dA
Sum of flux through all the elemental area gives rise to flux through the arbitrary surface.
\(int{dphi }=int{B.dA}\)If area vector is making angle \(overrightarrow{B}\) then \(int{dphi }=int{B.dA}\) cos Ɵ
Special cases:
Magnitude of flux is the function of three parameters they are strength of the magnetic induction\((overrightarrow{B})\), magnitude of area and angle (Ɵ) between the \(overrightarrow{B}\) and \(overrightarrow{A}\) with respect to this flux can be
φ = +ve and -ve
φ = 0 (can be zero also)
(i) Ɵ = 0° [Ɵ – Angle between \(overrightarrow{B}\) and\(overrightarrow{A}\)] \(begin{matrix} phi =B.cos theta ,,,,,,,,,,,,,,,,,,,,,, \ phi =BA,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, \ phi =max imum(theta =0{}^circ ) \ end{matrix}\)
(ii) Ɵ = 90°
φ = BAcos Ɵ
Magnetic flux density
Magnetic flux is the measure of intensity (or) strength field in the given area.
Magnetic flux is measured in unit is tesla – m2 = Weber
1Wb = Tm2
\(frac{Wb}{{{m}^{2}}}=1T\)Wb/m2 is the unit of magnetic induction \((overrightarrow{B})\) some time which is referred as magnetic flux density.
Magnetic flux density = is nothing strength of the magnetic induction.