An iron core is inserted into a solenoid 0.5 m long with 400 turns per unit length. The area of cross-section of the solenoid is 0.001 m². (a) Find the permeability of the core when a current of 5 A flows through the solenoid winding. Under these conditions, the magnetic flux through the cross-section of the solenoid is 1.6 x 10^–3 Wb. (b) Find the inductance of the solenoid under these conditions.

Solution

Given,
Length of the solenoid, l = 0.5 m
Number of turns per unit length, n = 400
Area of cross- section of the solenoid, A= 0.001 m²

The magnetic induction on the axis of the solenoid is given by

B = μ [μ_{0} nI] B = μ (\frac{μ_{0}}{4 π}) 4 πni . . . . (1)

where,
μ is the permeability of the medium,
n the number of turns per unit length and,
i is the current passing through the coil.

(a) The normal to the area is along the direction of the field.

Therefore,

Magnetic flux,

ϕ = BA,

Given,

Flux, ϕ = 1.6 \times 10^{- 3} Wb, A r e a, A = 0.001 = 10^{- 3} m^{2}

Therefore,

B = \frac{ϕ}{A} = \frac{1.6 \times 10^{- 3}}{10^{- 3}} = 1.6 Wb m^{- 2}

Since,

No . of turns, n = 400, Current, i = 5 A and, \frac{μ_{0}}{4 π} = 10^{- 7} {Hm}^{- 1}

We have form equation (1),

1.6 = μ \times 10^{- 7} \times 4 π \times 400 \times 5

which gives,

μ = \frac{1.6 \times 10^{7}}{4 π \times 5 \times 400} = 636.7 \approx 637

(b) Total number of turns in the solenoid is given by

N = n \times l = 400 \times 0.5 = 200

Total flux through the solenoid is,

N ϕ = 200 \times 1.6 \times 10^{- 3} = 0.32 Wb

Self inductance,

L = \frac{Nϕ}{i} = \frac{0.32}{5} = 0.064 H = 64 mH

Alternatively,

L = \frac{{μμ}_{0} N^{2} A}{l} = {μμ}_{0} n^{2} lA

= 637 \times (4 π \times 10^{- 7}) \times 400 \times 400 \times 0.5 \times 10^{- 3} = 6.4 \times 10^{- 2} H = 64 mH

Some More Questions From Moving Charges And Magnetism Chapter

Two moving coil meters, M₁ and M₂ have the following particulars:
R₁ = 10 Ω N₁ = 30,
A₁ = 3.6 x 10^–3 m², B₁ = 0.25 T
R₂ = 14 Ω; N₂ = 42,
A₂ = 1.8 x 10^–3 m², B₂ = 0.50 T
(The spring constants k are identical for the two meters).
Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M₂ and M_1.

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