A Small Loop Of Area 8 Cm² Inside A Long Solenoid

A Small Loop Of Area 8cm2 Is Placed Inside a Long Solenoid That Has 8

A small loop of area 8 cm2 is placed inside a long solenoid with 850 turns per meter that carries a sinusoidally varying current given by: i(t) = I0 sin(ωt), where I0 = 1.2 A is the amplitude and ω = 200 rad/s is the angular frequency. The central axis of the loop and the solenoid coincide. The task is to analyze the magnetic field, flux, inductance, and induced emf related to this setup.

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Understanding the electromagnetic phenomena associated with a solenoid and a co-axial loop involves analyzing magnetic fields, flux, inductance, and induced emf. This detailed analysis encompasses the calculation of the magnetic field inside the solenoid, the magnetic flux through the loop per unit length, the inductance per unit length of the solenoid, and the emf induced within the small loop due to the time-varying magnetic field.

1. Magnetic Field Inside the Solenoid, B(t)

The magnetic field within an ideal, long solenoid with N turns per unit length, carrying a current i(t), is given by:

B(t) = μ0 N i(t)

where μ0 = 4π × 10-7 T·m/A (permeability of free space). The number of turns per unit length, N, is given as 850 turns/m, and the current i(t) is sinusoidal with amplitude I0 = 1.2 A and ω = 200 rad/s:

i(t) = I0 sin(ωt)

Substituting the known values:

B(t) = (4π × 10-7) × 850 × 1.2 × sin(ωt)

Calculating the constants:

  • μ0 = 4π × 10-7 ≈ 1.2566 × 10-6 T·m/A
  • μ0 × N × I0 ≈ 1.2566 × 10-6 × 850 × 1.2 ≈ 1.2806 × 10-3 T

Thus, the magnetic field as a function of time is:

B(t) ≈ 1.2806 × 10-3 × sin(200 t) Т

where T represents Tesla, the unit of magnetic flux density.

2. Magnetic Flux per Unit Length Through the Windings, Φ(t)/l

The magnetic flux through the small loop, aligned with the solenoid axis, is given by:

Φ(t) = B(t) × A

where A = 8 cm2 = 8 × 10-4 m2 is the area of the loop. The flux per unit length of the solenoid (since the flux is uniform along its length) is:

Φ(t)/l = B(t) × A / l

But since B(t) is uniform across the cross-section and the length of the solenoid can be taken as unity for flux per unit length, we directly analyze in terms of flux per unit length as:

Flux per unit length (Φ(t)/l) = B(t) × A

Inserting the value for B(t):

Φ(t)/l ≈ 1.2806 × 10-3 × sin(200 t) × 8 × 10-4 = 1.0245 × 10-6 × sin(200 t) Weber/m

3. Inductance per Unit Length, L'

The inductance per unit length for a long solenoid is a well-established expression, derived from magnetic energy considerations:

L' = μ0 × N2

Given N = 850 turns/m:

Calculating:

  • L' = 1.2566 × 10-6 × (850)2 = 1.2566 × 10-6 × 722,500 ≈ 0.908 Henries per meter (H/m)

This implies that the inductance per unit length of the solenoid is approximately 0.908 H/m.

4. EMF Induced in the Small Loop

The emf induced in the small loop, according to Faraday’s Law, is proportional to the time derivative of the magnetic flux:

ε(t) = - dΦ(t)/dt

Calculating the derivative:

ε(t) = -d/dt [B(t) × A] = -A × dB(t)/dt

Since B(t) = Bmax sin(ωt), then:

dB(t)/dt = Bmax ω cos(ωt)

where Bmax ≈ 1.2806 × 10-3 T. Therefore, the emf is:

ε(t) ≈ -A × Bmax × ω × cos(ωt)

Substituting known values:

  • A = 8 × 10-4 m2
  • Bmax ≈ 1.2806 × 10-3 T
  • ω = 200 rad/s

The maximum emf amplitude (εmax) is:

εmax = A × Bmax × ω = 8 × 10-4 × 1.2806 × 10-3 × 200 ≈ 2.048 × 10-4 V

Thus, the induced emf varies sinusoidally as:

ε(t) ≈ -2.048 × 10-4 × cos(200 t) Volts

Summary and Conclusions

This analysis demonstrates how a sinusoidally varying current in a long solenoid generates a time-varying magnetic field, which in turn induces a fluctuating magnetic flux through a co-axial small loop. The magnetic field inside the solenoid reaches a maximum of approximately 1.28 millitesla, with a magnetic flux of about 8.2 microwebers per meter length. The inductance per unit length is significant at about 0.908 henries per meter, highlighting the solenoid's ability to oppose changes in current. The emf induced in the small loop peaks at roughly 0.204 millivolts, with a phase difference indicating a cosine dependence relative to the original magnetic flux variation. These findings are consistent with the principles of electromagnetic induction and the behavior of AC circuits in inductive systems.

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