If the loop is rotating at a constant angular velocity ω, then the angle θ of the loop will increase linearly with time.
θ = ωt
also, the tangential velocity v of the edges of the loop is:
v= r ω
where r is the radius from axis of rotation out to the edge of the loop and ω is the angular velocity of the loop.
= (v x B) . l
eind = 2r ωBl sin ωt
since area, A = 2rl,
= ABω sin ωt
Finally, since maximum flux through the loop occurs when the loop is perpendicular to the magnetic flux density lines, so
ω sin ωt
From here we may conclude that the induced voltage is dependent upon:
• Flux level (the B component)
• Speed of Rotation (the v component)
• Loop Constants (the l component and field materials)
The path of integration in Ampere’s law is the mean path length of the core, lc. The current passing within the path of integration Inet is then Ni, since the coil of wires cuts the path of integration N times while carrying the current i. Hence Ampere’s Law becomes,
In this sense, H (Ampere turns per metre) is known as the effort required to induce a magnetic field. The strength of the magnetic field flux produced in the core also depends on the material of the core. Thus,
B = μH
B = magnetic flux density (webers per square meter, Tesla (T))
µ= magnetic permeability of material (Henrys per meter)
H = magnetic field intensity (ampere-turns per meter)
The constant μ may be further expanded to include relative permeability which can be defined as below:
– permeability of free space (a.k.a. air)
Hence the permeability value is a combination of the relative permeability and the permeability of free space. The value of relative permeability is dependent upon the type of material used. The higher the amount permeability, the higher the amount of flux induced in the core. Relative permeability is a convenient way to compare the magnetizability of materials.
Also, because the permeability of iron is so much higher than that of air, the majority of the flux in an iron core remains inside the core instead of travelling through the surrounding air, which has lower permeability. The small leakage flux that does leave the iron core is important in determining the flux linkages between coils and the self-inductances of coils in transformers and motors.
B = μH = μNi
Now, to measure the total flux flowing in the ferromagnetic core, consideration has to be made in terms of its cross sectional area (CSA). Therefore,
φ = ∫BdA
Where: A – cross sectional area throughout the core
Assuming that the flux density in the ferromagnetic core is constant throughout hence constant A, the equation simplifies to be:
φ = BA
Taking into account past derivation of B,
φ = μNi