1. A closed magnetic circuit of cast steel contains a 6cm long path of cross sectional area 1cm square and a 2cm part of cross sectional area of 0.5cm square. A coil of 200 turns is wound around the 6cm length of the circuit and a current of 0.4ampere flows. Determine the flux density in the 2cm path if the relative permeability of cast steel is 750.
2. A flux of 25m wb links of a 1500turns coil when a current of 3ampere passes through the coil, calculate
i. The inductance of the coil
ii. The average emf induced if the current falls to 0 in 15ms.
1. To determine the flux density in the 2cm path, we can use the formula:
Flux density (B) = (μ * N * I) / (A * L)
Where:
B = Flux density
μ = Relative permeability (750 for cast steel)
N = Number of turns (200)
I = Current (0.4 Ampere)
A = Cross-sectional area (0.5cm square)
L = Length of the path (2cm)
Substituting the given values into the formula:
B = (750 * 200 * 0.4) / (0.5 * 2)
B = 60000 / 1
B = 60000 Gauss
Therefore, the flux density in the 2cm path is 60000 Gauss.
2. To calculate the inductance of the coil, we can use the formula:
Inductance (L) = (Φ / I)
Where:
L = Inductance
Φ = Flux (25m wb)
I = Current (3 Ampere)
Substituting the given values into the formula:
L = (25 * 10^-3) / 3
L ≈ 0.0083 Henrys
Therefore, the inductance of the coil is approximately 0.0083 Henrys.
To calculate the average emf induced when the current falls to 0 in 15ms, we can use the formula:
Average emf = -(ΔΦ / Δt)
Where:
ΔΦ = Change in Flux (25m wb - 0)
Δt = Change in time (15ms)
Substituting the given values into the formula:
Average emf = -((25 * 10^-3) - 0) / (15 * 10^-3)
Average emf = - (25 * 10^-3) / (15 * 10^-3)
Average emf ≈ -1.67 Volts
Therefore, the average emf induced when the current falls to 0 in 15ms is approximately -1.67 Volts.
To answer the questions, we need to use the formulas related to magnetic circuits and electromagnetic induction.
1. To determine the flux density in the 2cm path of the magnetic circuit, we can use the formula:
B = (μ * N * I) / (A * l)
where:
B is the magnetic flux density
μ is the relative permeability
N is the number of turns
I is the current flowing through the coil
A is the cross-sectional area of the path
l is the length of the path
In this case, the length of the path in the 6cm part is 6cm (0.06m), and the length of the path in the 2cm part is 2cm (0.02m). The cross-sectional area of the 6cm part is 1cm^2 (0.0001m^2), and the cross-sectional area of the 2cm part is 0.5cm^2 (0.00005m^2). The number of turns is 200, and the current is 0.4 Ampere. The relative permeability of cast steel is given as 750.
For the 6cm part of the circuit:
B1 = (μ * N * I) / (A1 * l1)
B1 = (750 * 200 * 0.4) / (0.0001 * 0.06)
B1 ≈ 8,000,000 Amperes per meter (A/m)
For the 2cm part of the circuit:
B2 = (μ * N * I) / (A2 * l2)
B2 = (750 * 200 * 0.4) / (0.00005 * 0.02)
B2 ≈ 48,000,000 Amperes per meter (A/m)
Therefore, the flux density in the 2cm path is approximately 48,000,000 A/m.
2. i. To calculate the inductance of the coil, we can use the formula:
L = (N^2 * Φ) / I
where:
L is the inductance of the coil
N is the number of turns
Φ is the magnetic flux linking the coil
I is the current passing through the coil
In this case, the number of turns is 1500, and the magnetic flux Φ is 25 mWb (convert it to Weber by dividing by 1000 to get 0.025 Wb). The current passing through the coil is 3 Ampere.
L = (1500^2 * 0.025) / 3
L ≈ 12,500 Henrys (H)
Therefore, the inductance of the coil is approximately 12,500 H.
2. ii. To calculate the average EMF (electromotive force) induced when the current falls to 0 in 15 ms, we can use the formula:
EMF = -L * (ΔI / Δt)
where:
EMF is the average electromotive force induced
L is the inductance of the coil
ΔI is the change in current (from 3A to 0A)
Δt is the change in time (15 ms, converted to seconds by dividing by 1000 to get 0.015 s)
In this case, the change in current is 3A - 0A = 3A, and the change in time is 0.015s.
EMF = -12,500 * (3 / 0.015)
EMF ≈ -2,500,000 Volts (V)
Therefore, the average EMF induced when the current falls to 0 in 15 ms is approximately -2,500,000 V. The negative sign indicates the direction of the induced EMF.