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A square coil is a loop of wire shaped in a perfect square. One of the key features of a square coil is that all four sides are equal in length. When considering a coil made from a single, fixed length of wire, like in the exercise, we can easily find each side's length. You can calculate each side of a square coil by dividing the total length of the wire by four. This is because a square has four equal sides. For example, if the total length is denoted as \( L \), each side of the square is \( s = \frac{L}{4} \).

This simplicity helps us especially when it comes to calculating the coil's area, which is essential in determining the magnetic properties of the coil. The area \( A \) of a square is given by the formula \( A = s^2 \). Hence for our square coil, the area is \( \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} \).

When this coil is placed in a magnetic field and carries current, it experiences magnetic torque. This is a turning force acting on the coil, explained further in the upcoming sections.

A rectangular coil is a loop of wire bent into a rectangle, which means it has a pair of longer sides and a pair of shorter sides. In the context of our exercise, the rectangular coil is crafted from the same length of wire as the square coil.

The challenge comes in distributing the wire to accommodate the rectangle's shape. Given that the longer side is twice the length of the short side, it's necessary to solve for each side's length using the total length of wire \( L \). Solving \( 2a + 2(2a) = L \) leads to \( a = \frac{L}{6} \) for the short side, and the long side is \( 2a = \frac{L}{3} \).

The area of the rectangle \( A \) is derived from multiplying its sides, \( a \times 2a = \frac{L}{6} \times \frac{L}{3} = \frac{L^2}{18} \).

Having a different shape from the square coil, the rectangular coil exhibits different properties when subjected to a magnetic field, especially concerning torque, as will be explored.

When a square or rectangular coil carrying current is placed within a magnetic field, it experiences a force called magnetic torque. The magnetic torque \( \tau \) is a result of the interaction between the magnetic field and the electric current flowing through the coil.

This torque is governed by the formula: \( \tau = nIAB \sin \theta \), where:

\( n \) is the number of turns in the coil

\( I \) is the current passing through the coil

\( A \) is the coil's area

\( B \) is the strength of the magnetic field

\( \theta \) is the angle between the magnetic field direction and the normal to the coil's plane

For maximum torque, the angle \( \theta \) is set to 90 degrees, making \( \sin \theta = 1 \). This means the plane of the coil is perpendicular to the magnetic field for maximal torque, maximizing the coil’s ability to rotate or twist.

In physics, comparing the capabilities of different structures in similar conditions is important for understanding their efficiency. In our exercise, we compare the maximum torque experienced by both the square and rectangular coils.

Since the formula \( \tau = IAB \) (for maximum torque where \( \sin \theta = 1 \)) involves the areas \( A \) of the respective coils under a constant magnetic field and current, the maximum torque experienced by each coil depends on these areas. For the square coil, the maximum torque is \( I \cdot \frac{L^2}{16} \cdot B \), while for the rectangular coil, it’s \( I \cdot \frac{L^2}{18} \cdot B \).

To find the ratio of their maximum torques, divide the expression for the square coil’s torque by the rectangular coil's torque, resulting in \( \frac{\tau_{\text{square}}}{\tau_{\text{rectangle}}} = \frac{\frac{L^2}{16}}{\frac{L^2}{18}} = \frac{18}{16} = \frac{9}{8} \).

This ratio \( \frac{9}{8} \) signifies that the square coil experiences a slightly higher maximum torque compared to the rectangular coil when exposed to identical conditions in a magnetic field.