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)

jupiter is the largest planet in our solar system, having a mass and radius that are respectively, 318 and 11.2 times that of earth. suppose that an object falls from rest near the surface of each planet and that the acceleration due to gravity remains constant during the fall. each object falls the same distance before striking the ground. determine the ratio of the time of fall on jupiter to that on earth.

i think the answer is a and b

i wish you luck