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

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)