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)

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