You would like to store 8.1 J of energy in the magnetic field of a solenoid. The solenoid has 620 circular turns of diameter 6.6 cm distribu

Question

You would like to store 8.1 J of energy in the magnetic field of a solenoid. The solenoid has 620 circular turns of diameter 6.6 cm distributed uniformly along its 33 cm length.
A. How much current is needed?
_____________ A
B. What is the magnitude of the magnetic field inside the solenoid?
________________T
C. What is the energy density (energy/volume) inside the solenoid?
________________ kJ/m^3

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Ngọc Khuê 3 months 2021-08-13T20:55:27+00:00 1 Answers 0 views 0

Answers ( )

    0
    2021-08-13T20:57:02+00:00

    Answer:

    (a) The current needed is 56.92 A

    (b) The magnitude of the magnetic field inside the solenoid is 0.134 T

    (c) The energy density inside the solenoid is 7.144 kJ/m³

    Explanation:

    Given;

    energy stored in the magnetic field of solenoid, E = 8.1 J

    number of turns of the solenoid, N = 620 turns

    diameter of the solenoid, D = 6.6 cm = 0.066 m

    radius of the solenoid, r = D/2 = 0.033 m

    length of the solenoid, L = 33 cm = 0.33 m

    Inductance of the solenoid is given as;

    L= \frac{\mu_o N^2 A}{l}

    where;

    A is the area of the solenoid = πr² = π (0.033)² = 0.00342 m²

    μ₀ is permeability of free space = 4π x 10⁻⁷ H/m

    L= \frac{4\pi*10^{-7} *620^2 *0.00342}{0.33} \\\\L = 0.005 \ H

    (A). How much current needed

    Energy stored in magnetic field of solenoid is given as;

    E = \frac{1}{2} LI^2\\\\

    Where;

    I is the current in the solenoid

    E = \frac{1}{2} LI^2\\\\I^2 = \frac{2E}{L}\\\\I = \sqrt{\frac{2*8.1}{0.005}}\\\\ I = 56.92 \ A

    (B) The magnitude of the magnetic field inside the solenoid

    B = μ₀nI

    where;

    n is number of turns per unit length

    B = μ₀(N/L)I

    B = (4π x 10⁻⁷)(620/0.33)(56.92)

    B = 0.134 T

    (C) The energy density (energy/volume) inside the solenoid

    U_B = \frac{B^2}{2\mu_0} \\\\U_B = \frac{(0.134)^2}{2*4\pi*10^{-7}} \\\\U_B = 7143.54 \ J/m^3\\\\U_B = 7.144 \ kJ/m^3

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