A magnetic dipole with a dipole moment of magnitude 0.0243 J/T is released from rest in a uniform magnetic field of magnitude 57.5 mT. The rotation of the dipole due to the magnetic force on it is unimpeded. When the dipole rotates through the orientation where its dipole moment is aligned with the magnetic field, its kinetic energy is 0.458 mJ. (a) What is the initial angle between the dipole moment and the magnetic field

47.76°

Explanation:

Magnitude of dipole moment = 0.0243J/T

Magnetic Field = 57.5mT

kinetic energy = 0.458mJ

∇U = -∇K

Uf - Ui = -0.458mJ

Ui - Uf = 0.458mJ

(-μBcosθi) - (-μBcosθf) = 0.458mJ

rearranging the equation,

(μBcosθf) - (μBcosθi) = 0.458mJ

μB * (cosθf - cosθi) = 0.458mJ

θf is at 0° because the dipole moment is aligned with the magnetic field.

μB * (cos 0 - cos θi) = 0.458mJ

but cos 0 = 1

(0.0243 * 0.0575) (1 - cos θi) = 0.458*10⁻³

1 - cos θi = 0.458*10⁻³ / 1.397*10⁻³

1 - cos θi = 0.3278

collect like terms

cosθi = 0.6722

θ = cos⁻ 0.6722

θ = 47.76°

The initial angle between the dipole moment and the magnetic field is 47.76⁰

Explanation:

Given;

magnitude of dipole moment, μ = 0.0243 J/T

magnitude of magnetic field, B = 57.5 mT

change in kinetic energy, ΔKE = 0.458 mJ

ΔKE = - ΔU

ΔKE = - (U₂ -U₁)

ΔKE = U₁ - U₂

U₁ -U₂ = 0.458 mJ

where;

θ₁ is the initial angle between the dipole moment and the magnetic field

is the final angle which is zero (0) since the  dipole moment is aligned with the magnetic field

μB(cos0 - cosθ₁) = 0.458 mJ

Substitute the given values of μ and B

0.0243 x 0.0575 (1 - cosθ₁) = 0.000458

0.00139725 (1 - cosθ₁) = 0.000458

(1 - cosθ₁) = 0.000458 / 0.00139725

(1 - cosθ₁) = 0.327787

cosθ₁ = 1 -  0.327787

cosθ₁ = 0.672213

θ₁ = cos⁻¹ (0.672213)

θ₁ = 47.76⁰

Thus, the initial angle between the dipole moment and the magnetic field is 47.76⁰

when jerome paddle in flowing river towards east the net speed of jerome will increase

so the net speed of jerome will be given as

now the speed of paul will be same as it is given as he is running on the ground

so

so here we can say since the speed of jerome is more so he will win the race

now as we know the distance so the time difference is given as

so jerome will reach first and the time difference will be 203.55 s or 3.4 min