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23 August, 05:47

A circular loop of wire with area A lies in the xy-plane. As viewed along the z-axis looking in the - z-direction toward the origin, a current I is circulating clockwise around the loop. The torque produced by an external magnetic field B⃗ is given by τ⃗ = D (2i^-4j^), where D is a positive constant, and for this orientation of the loop the magnetic potential energy U=-μ⃗ ⋅B⃗ is negative. The magnitude of the magnetic field is B0=15D/IA.

Determine the vector magnetic moment of the current loop.

Express your answer in terms of the variables I I, A A, i^ / hat i, j^ / hat j, and k^ / hat k.

Determine the component Bx B_x of B? / vec B.

Determine the component By B_y of B

Determine the component Bz B_z of B

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  1. 23 August, 06:07
    0
    A) μ^ = - IA•k

    B) Bx = 2D/IA

    C) By = 4D/IA

    D) Bz = - 14D / (IA)

    Step-by-step explanation:

    We are given;

    Torque; τ = D (2i^ - 4j^) Nm

    Potential energy; U = - μ•B

    Magnitude of magnetic field;

    Bo = 15D/IA

    a. The vector magnetic moment of the current loop is given as

    μ^ = - μ•k

    μ^ = - IA •k

    b. Now, let's find the component of the magnetic field B.

    If we assume B = Bx•i + By•j + Bz•k

    Then, torque is given as

    τ = μ^ * B

    τ = - IA •k * (Bx•i + By•j + Bz•k)

    Note that;

    i*i=j*j*k*k=0

    i*j=k. j*i=-k

    j*k=i. k*j=-i

    k*i=j. i*k=-j

    Then,

    τ = - IA •k * (Bx •i + By •j + Bz •k)

    τ = - IABx• (k*i) - IABy• (k*j) - IABz• (k*k)

    τ = - IABx•j + IABy•i

    τ = IABy•i - IABx•j

    The given torque is τ = D (2i^ - 4j^)

    Comparing coefficients;

    Then,

    -IABx = - 4D

    Bx = - 4D/-IA

    Bx = 4D/IA

    c. Also,

    IABy = 2D

    Then, By = 2D/IA

    d. To get Bz, let's use the magnitude of magnetic field Bo

    Bo² = Bx² + By² + Bz²

    (15D/IA) ² = (4D/IA) ² + (2D/IA) ² + Bz²

    Bz² = (15D/IA) ² - (4D/IA) ² - (2D/IA) ²

    Bz² = 225D² / (I²A²) - 16D² / (I²A²) - 4D² / (I²A²)

    Bz² = (225D² - 16D² - 4D²) / I²A²

    Bz² = 205D²/I²A²

    Bz = √ (205D² / (I²A²))

    Bz = ± 14D / (IA)

    So we want to determine if Bz is positive or negative

    From the electric potential,

    U = - μ•B

    U = - ( - IA k• (Bx i+By j+Bz k)

    Note, - * - = +, i. i=j. j=k. k=1

    i. j=j. k=k. i=0

    Then,

    U = IA k• (Bx i+By j+Bz k)

    U = IABz

    Since we are told that U is negative, then this implies that Bz is negative

    Then, Bz = - 14D / (IA)
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