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(5i^−5j^), 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=14D/IA.

A) Determine the vector magnetic moment of the current loop.

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

B) Determine the component Bx of B⃗ .

Express your answer using one significant figure.

C) Determine the component By of B⃗ .

Express your answer using one significant figure.

D) Determine the component Bz of B⃗ .

Express your answer using two significant figures.

a. μ~ = - IA •k

b. Bx = 5D/IA

c. By = 5D/IA

d. Bz = -12.1D/IA

Explanation:

Given that,

Torque, τ = D(5i^ − 5j^) Nm

Potential energy is gvwn as

U=− μ•B

Magnitude of magnetic field is

Bo=14D/IA

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

μ~ = - μ•k

μ~ = - IA •k

b. Now to find the component of the magnetic field B.

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(5i^ − 5j^)

Comparing coefficient

Then,

IABy=5D

Then, By= 5D/IA

c. Also,

-IABx=-5D

Bx=-5D/-IA

Bx=5D/IA

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

Bo²=Bx²+By²+Bz²

(14D/IA)²=(5D/IA)²+(5D/IA)² + Bz²

Bz²=(14D/IA)²- (5D/IA)²-(5D/IA)²

Bz²=196D²/I²A²-25D²/I²A²-25D²/I²A²

Bz²=(196D²-25D²-25D²)/I²A²

Bz²=146D²/I²A²

Bz=√(146D²/I²A²)

Bz=± 12.1 D/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= -12.1D/IA

A) Vector μ = -IA•k^

B) Bx = 5D/IA

C) By = 5D/IA

D) Bz = -12D/IA

Explanation:

I have attached the explanation for ease of understanding

the order for strong to weak forces such as

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

the fundamental forces are those forces that act between to objects or element to make interactions. there are four types of fundamental forces such as strong and weak nuclear forces (due to protons and electrons), electromagnetic force (due to the motion of electrons) and gravitational force (due to orbital movement of the planet). if these forces are arranged on the basis of their maximum strength then the order is

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

snell's law

refractve index = sin incidence / sin refraction