Answer:
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)
