A coil of N turns is wrapped around an iron ring of radius ‘d’ and cross-section
‘A’ (d > > A). Assuming a constant permeability, mu > > 1 for the iron.
Find the following :
(i) the magnetic flux as a function of the current ‘i’, and
(ii) the magnetic flux if a gap of width delta ^2 < < A) is cut in the ring.
To find the magnetic flux as a function of the current 'i' in the coil wrapped around the iron ring, you can use the formula for magnetic flux through a coil:
Φ = N * B * A
Where:
- Φ is the magnetic flux
- N is the number of turns in the coil
- B is the magnetic field strength
- A is the cross-sectional area of the coil
In this case, assuming a constant permeability μ >> 1 for the iron, we can use the formula for magnetic field strength inside a solenoid:
B = μ * N * i / L
Where:
- i is the current in the coil
- L is the length of the coil
Since the coil is wrapped around the iron ring, we can assume that L is approximately equal to the circumference of the ring, which is 2πd.
Substituting this into the previous formula, we get:
B = μ * N * i / (2πd)
Now we can substitute this value of B into the formula for magnetic flux:
Φ = N * B * A
Φ = N * (μ * N * i / (2πd)) * A
Simplifying further, we get:
Φ = (μ * N^2 * A * i) / (2πd)
This gives us the magnetic flux as a function of the current 'i' in the coil wrapped around the iron ring.
Next, to find the magnetic flux if a gap of width Δ^2 (where Δ^2 << A) is cut in the ring, we need to consider the effect of the gap on the flux.
When a gap is introduced, the magnetic field lines passing through the gap reduce, resulting in a decrease in the magnetic flux.
To calculate the flux in this case, we can consider the area of the ring with the gap as the new effective cross-sectional area, which is A - Δ^2.
Now, we can substitute this value into the previous formula for magnetic flux:
Φ = (μ * N^2 * (A - Δ^2) * i) / (2πd)
This gives us the magnetic flux if a gap of width Δ^2 (where Δ^2 << A) is cut in the iron ring.