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A very long, straight solenoid with a cross-sectional area of 1.90 cm2 is wound with 86.6 turns of wire per centimeter. Starting at t = 0, the current in the solenoid is increasing according to i(t)= ( 0.176 A/s2 )t2. A secondary winding of 5.0 turns encircles the solenoid at its center, such that the secondary winding has the same cross-sectional area as the solenoid. What is the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 3.2A ?

Respuesta :

davidlcastiblanco

Answer:

|Emf| = 1.58x10⁻⁵ V

Explanation:

N = 88.6 turns/cm

N = 8860 turns/metre

The field inside the long solenoid is given by

B= μ₀Ni

B=  4πx10⁻⁷ x 8860 x 0.1761t² =

B= 1.96x10⁻³ t²

dB/dt = 3.92x10⁻³ t

A = 1.90cm² = 1.90x10⁻⁴ m²

|Emf| = rate of change of flux linkage

|Emf| = d(NAB)/dt

|Emf|= NA dB/dt

|Emf|= 5 x 1.90x10⁻⁴* 3.92x10⁻³ t

|Emf|= 3.724x10⁻⁶ t

if T is the time at which the current = 3.2A

3.2 A = 0.176T²

T = 4.26s

|Emf|=  3.724x10⁻⁶t *4.26s

|Emf| = 1.58x10⁻⁵ V

onyebuchinnaji

The magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 1.55 x 10⁻⁵ V.

The given parameters;

  • area of the wire, A = 1.9 cm²
  • number of turns of turns of the wire, N = 86.6 turns/cm = 8660 turns/m
  • current in the solenoid, I = 3.2 A
  • secondary number of turns, = 5

The magnetic field at the center of the solenoid is calculated as follows;

[tex]B = \mu_o nI\\\\B = (4\pi \times 10^{-7}) \times (8660) \times (0.176)t^2\\\\B = 0.001916 \ t^2\\\\\frac{dB}{dt} = 0.00383 \ t[/tex]

The induced emf in secondary winding is calculated as follows;

[tex]emf =N \frac{dB}{dt} A\\\\emf = 5 \times 0.00383\times 1.9 \times 10^{-4}\\\\emf = 3.64 \times 10^{-6} t \[/tex]

from the given current;

[tex]3.2 = 0.176t^2\\\\t^2 = \frac{3.2}{0.176} \\\\t = \sqrt{\frac{3.2}{0.176}} \\\\t = 4.26 \ s[/tex]

The induced emf is calculated as follows;

[tex]emf = 3.64 \times 10^{-6} \times 4.26 \\\\emf = 1.55 \times 10^{-5} \ V[/tex]

Thus, the magnitude of the emf induced in the secondary winding at the instant that the current in the solenoid is 1.55 x 10⁻⁵ V.

Learn more here:https://brainly.com/question/13744192

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