91Ó°ÊÓ

Magnetic fields are a fundamental concept in physics, providing forces that act on charged particles and current-carrying wires, like our coil. The strength of this field is represented by the symbol \( B \) and is measured in teslas (T). In our exercise, the given magnetic field strength is \( 0.75 \) T, which indicates the intensity of the magnetic field that the coil experiences. Understanding magnetic field strength is crucial, as it directly influences the force exerted on the coil.
The stronger the magnetic field, the greater the force, which translates into a larger torque on the coil. This is why magnetic field strength appears in the torque formula, \( \tau = n \times B \times I \times A \). In this scenario, a precise value of \( B \) helps in calculating other variables related to the coil, like its area or the torque it experiences.

The area of the coil, denoted by \( A \), plays a key role in determining the torque a coil experiences in a magnetic field. For a circular coil, this area depends on its radius. In our problem, we begin by finding the area from the known torque, which uses the formula \( \tau = n \times B \times I \times A \).
Given the torque \( \tau = 8.4 \times 10^{-4} \) N·m, and knowing \( n = 1 \), \( B = 0.75 \) T, and \( I = 3.7 \) A, the area \( A \) can be calculated as:

• Substituting the values into the torque formula.
• Rearranging gives \( A = \frac{8.4 \times 10^{-4}}{0.75 \times 3.7} \).
• Solving, we obtain \( A \approx 3.029 \times 10^{-4} \text{ m}^2 \).

Knowing the area helps to find the radius and ultimately the length of the wire, showcasing its significance in physics problems involving torque.

The current, denoted as \( I \), that flows through the coil is a major player in determining the torque the coil experiences in a magnetic field. In this context, the current is \( 3.7 \) amps (A). When we talk about current, we're referring to the flow of electric charge across the wire of the coil. This act of flowing charge interacts with the magnetic field to produce torque.
The formula for torque involves current as it directly affects how much force is experienced. More current means a larger magnetic force, enhancing the torque. This is seen in the equation \( \tau = n \times B \times I \times A \). Thus, understanding current not only helps in problem-solving but also provides a peek into how electrical and magnetic fields interact.

The geometry of the coil significantly impacts its physical and mathematical properties. Here, we're dealing with a circular coil, meaning it is shaped as a circle. A crucial feature of circular geometry is that it allows the area \( A \) to be expressed as \( \pi r^2 \), where \( r \) is the radius of the coil.
In our problem, once the area is calculated, we can find the radius:

• Set up the equation as \( \pi r^2 = 3.029 \times 10^{-4} \).
• This simplifies to \( r^2 = \frac{3.029 \times 10^{-4}}{\pi} \).
• Solving for the radius gives \( r \approx 9.81 \times 10^{-3} \text{ m} \).

Knowing the radius makes it possible to calculate the circumference \( C \), which represents the coil's length. This relationship explains why the geometry of the coil helps translate into practical measurements like wire length.