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A solenoid with 181 turns/cm and a diameter of 3.4 cm is coaxially placed within a 110-turn coil with a radius of 1.8 cm and a resistance of 6.2 Ω. If the current in the solenoid drops from [tex]2.0 \, A[/tex] to [tex]0 \, A[/tex] over a time interval of [tex]35 \, ms[/tex], what is the induced current in the coil during the same time interval?

Answer :

To determine the induced current in the coil, we need to understand the process of electromagnetic induction. Faraday's Law of Induction states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit.

Given:


  • Solenoid: 181 turns/cm

  • Diameter of the solenoid: 3.4 cm

  • Radius of the outer coil: 1.8 cm

  • Resistance of the outer coil: 6.2 [tex]\Omega[/tex]

  • Change in current in the solenoid: from 2.0 A to 0 A

  • Time interval: 35 ms (which is 0.035 seconds)


Steps to Solve:


  1. Calculate the length of the solenoid in meters:
    The number of turns per meter is [tex]18100[/tex] turns/m (since 1 cm = 0.01 m).


  2. Calculate the initial and final magnetic field inside the solenoid:
    The magnetic field [tex]B[/tex] of a solenoid is given by:
    [tex]B = \mu_0 n I[/tex]
    where:
    [tex]\mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}[/tex] (permeability of free space),
    [tex]n = 18100 \ \text{turns/m}[/tex],
    [tex]I = 2.0 \ \text{A}[/tex] initially and [tex]0 \ \text{A}[/tex] finally.


  3. Calculate the change in magnetic field ([tex]\Delta B[/tex]) inside the solenoid:
    [tex]\Delta B = \mu_0 \cdot n \cdot \Delta I[/tex]
    where [tex]\Delta I = 2.0 \ \text{A} - 0 \ \text{A} = 2.0 \ \text{A}[/tex].


  4. Calculate the EMF induced in the outer coil:
    Faraday’s law states:
    [tex]\epsilon = -N \cdot \frac{\Delta \Phi}{\Delta t}[/tex]
    The magnetic flux [tex]\Phi[/tex] through the outer coil is [tex]B \cdot A[/tex], where [tex]A[/tex] is the area of a circle with radius equal to the solenoid radius.
    Area [tex]A = \pi r^2[/tex], where [tex]r = \frac{3.4\ \text{cm}}{2} = 0.017\ \text{m}[/tex].


  5. Calculate the change in flux through the outer coil:
    [tex]\Delta \Phi = \Delta B \cdot \pi (0.017)^2[/tex]


  6. Induced EMF ([tex]\epsilon[/tex]) in the coil:
    Substituting the results into Faraday’s law:
    [tex]\epsilon = -110 \cdot \frac{\Delta B \cdot \pi (0.017)^2}{0.035}[/tex]


  7. Calculate the induced current in the outer coil.
    Using Ohm’s law, [tex]I = \frac{\epsilon}{R}[/tex], where [tex]R = 6.2 \ \Omega[/tex].


  8. Final Calculation: (Compute the values)


    • Substitute values to get [tex]\Delta B[/tex].

    • Calculate [tex]\Delta \Phi[/tex].

    • Solve the equation for [tex]\epsilon[/tex].

    • Compute [tex]I[/tex] using Ohm’s law.




Conclusion:

By following these steps and performing the calculations, you can find the exact value of the induced current in the outer coil.

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Rewritten by : Barada