Engineering

Physics

Magnetic Moment

Question

A person wants to roll a solid non-conducting spherical ball of mass m and radius r on a surface whose coefficient of static friction is µ. He placed the ball on the surface wrapped with n turns of closely packed conducting coils of negligible mass at the perimeter. By some arrangement he is able to pass a current i through the coils either in the clockwise direction or in the anticlockwise direction. A constant horizontal magnetic field $\vec{B}$ is present throughout the space as shown in the fig. (Assume µ is large enough to help rolling motion)

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Linked Question 1

If current i is passed through the coils the initial torque in the coil is

A

$- {πnir}^{2} B \hat{k}$

B

${πnir}^{2} B \hat{j}$

C

${πnir}^{2} B \hat{k}$

D

$- {πnir}^{2} B \hat{j}$

$\vec{τ} = \vec{M} \times \vec{B}; \vec{M} = {nπIr}^{2} \hat{j}; \vec{B} = B \hat{i}$

$\vec{τ} = {nπIr}^{2} B (- \hat{k})$

Linked Question 2

The angular velocity of the ball when it has rotated through an angle θ is (θ < 180°), is

A

$\sqrt{\frac{5}{7} \frac{πniB}{m} sinθ}$

B

$\sqrt{\frac{5}{14} \frac{πniB}{m} cosθ}$

C

$\sqrt{\frac{5}{14} \frac{πniB}{m} sinθ}$

D

$\sqrt{\frac{10}{7} \frac{πniB}{m} sinθ}$

${\vec{τ}}_{net} = | (\vec{M} \times \vec{B}) - \vec{f} \times \vec{R} | = \frac{2}{5} {MR}^{2} α$

$\Rightarrow \int_{0}^{θ} {nπIR}^{2} Bcosθdθ - f R = \frac{2}{5} {MR}^{2} α$

f = Ma = MRα

${nπIR}^{2} B sinθ = \frac{7}{5} {MR}^{2} ω \frac{ω^{2}}{2}$

$ω = \sqrt{\frac{10}{7} \frac{πniBsinθ}{m}}$