Moving Charges and Magnetism

A coil carrying a current experiences a torque when placed in a magnetic field. This torque is quantified by the following relation:$\tau = N \cdot I \cdot A \cdot B \cdot \sin \theta$where $N$ is the number of turns in the coil, $I$ is the current flowing through the coil, $A$ is the area of the coil, $B$ is the magnetic field strength, and $\theta$ is the angle between the magnetic field and the normal to the plane of the coil.The torque is at its maximum when $\sin \theta = 1$, which occurs at $\theta = 90^{\circ}$. Thus, the maximum torque $\tau_{\max}$ can be expressed as:$\tau_{\max} = N \cdot I \cdot A \cdot B$According to the problem, the experienced torque is $80\%$ of this maximum:$N \cdot I \cdot A \cdot B \cdot \sin \theta = 0.8 \cdot \tau_{\max}$Substituting $\tau_{\text{max}}$ from equation (i):$N \cdot I \cdot A \cdot B \cdot \sin \theta = \frac{80}{100} \cdot (N \cdot I \cdot A \cdot B)$This simplifies to:$\sin \theta = \frac{4}{5}$We know that:$\cos \theta = \sqrt{1 - \sin^2 \theta} = \frac{3}{5}$Thus, the tangent of the angle $\theta$ becomes:$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{4/5}{3/5} = \frac{4}{3}$Therefore, the angle $\theta$ is:$\theta = \tan^{-1} \left( \frac{4}{3} \right)$