Key Equations:

Magnetic field at center: \( B = \frac{\mu_0 NI}{2R} \)

Magnetic moment: \( m = NIA = NIπr^2 \)

Torque: \( \tau = mB\sinθ \)

Angular speed: \( \omega = \sqrt{\frac{2mB}{I}} \)

(a) Field at Center:

\[ B = \frac{\mu_0 NI}{2R} \]

Given: \( N=100 \), \( I=3.2 \) A, \( R=0.1 \) m

\[ B = \frac{4π×10^{-7}×100×3.2}{2×0.1} = 2×10^{-3} \text{ T} \]

Direction given by right-hand thumb rule.

(b) Magnetic Moment:

\[ m = NIA = NIπr^2 \]

\[ m = 100×3.2×3.14×10^{-2} = 10 \text{ A·m}^2 \]

Direction given by right-hand thumb rule.

(c) Torques:

\[ \tau = mB\sinθ \]

Initial (θ=0°): \( \tau_i = 10×2×\sin0° = 0 \text{ N·m} \)

Final (θ=90°): \( \tau_f = 10×2×\sin90° = 20 \text{ N·m} \)

(d) Angular Speed:

From work-energy principle:

\[ \frac{1}{2}I\omega^2 = mB(1-\cos90°) \]

\[ \frac{1}{2}×0.1×\omega^2 = 10×2×1 \]

\[ \omega = \sqrt{\frac{2×20}{0.1}} = 20 \text{ rad/s} \]