A circular wire of radius \(5.0 \mathrm{~cm}\) has a current of \(3.0 \mathrm{~A}\) flowing in it. The wire is placed in a uniform magnetic field of \(5.0 \mathrm{mT}.\) a) Determine the maximum torque on the wire. b) Determine the range of the magnetic potential energy of the wire.

The area of a circle is given by the formula: \(A = \pi * r^2\) where \(r\) is the radius of the circle. In this case, the radius is \(5.0 \mathrm{~cm} = 0.05 \mathrm{~m}\). So, \(A = \pi * (0.05)^2 = 0.007854 \mathrm{~m^2}\).

The maximum torque on a closed current loop in a magnetic field is given by the formula: \(\tau = I * A * B * sin(\theta)\) where \(I\) is the current, \(A\) is the loop area, \(B\) is the magnetic field, and \(\theta\) is the angle between the normal to the plane of the loop and the magnetic field. The maximum torque occurs when the angle \(\theta = 90°\), so \(sin(\theta) = 1\). Substituting the given values and the calculated area, \(\tau = (3.0 \mathrm{~A})(0.007854 \mathrm{~m^2})(5.0 * 10^{-3}\mathrm{~T})(1)= 0.00011781 \mathrm{~N\cdot m}\) So, the maximum torque on the wire is \(0.00011781 \mathrm{~N\cdot m}\).

The magnetic potential energy of a closed current loop in a magnetic field is given by the formula: \(U = -\tau * cos(\theta)\) where \(\tau\) is the torque and \(\theta\) is the angle between the normal to the plane of the loop and the magnetic field. For minimum potential energy, \(\theta = 0°\) (loop normal is parallel to the field), so \(cos(\theta) = 1\). Therefore, \(U_\text{min} = -\tau * cos(0°)\) \(U_\text{min} = -(0.00011781 \mathrm{~N\cdot m})(1) = -0.00011781 \mathrm{~J}\) For maximum potential energy, \(\theta = 180°\) (loop normal is anti-parallel to the field), so \(cos(\theta) = -1\). Therefore, \(U_\text{max} = -\tau * cos(180°)\) \(U_\text{max} = -(0.00011781 \mathrm{~N\cdot m})(-1) = 0.00011781 \mathrm{~J}\) The range of the magnetic potential energy of the wire is from \(-0.00011781 \mathrm{~J}\) to \(0.00011781 \mathrm{~J}\).