Problem 36 - Physics

Question

A wire has a length of \(7.00 \times 10^{-2} \mathrm{~m}\) and is used to make a circular coil of one turn. There is a current of \(4.30 \mathrm{~A}\) in the wire. In the presence of a 2.50-T magnetic field, what is the maximum torque that this coil can experience?

Step-by-Step Solution

Answer

The maximum torque is approximately \( 4.17 \times 10^{-3} \mathrm{~Nm} \).

Step 1: Determine the Radius of the Coil

The wire makes one turn forming a circular coil. The length of the wire is the circumference of the circle, which can be calculated using the formula:\[ C = 2\pi r \]where \( C \) is the circumference (7.00 \times 10^{-2} \mathrm{~m}), and \( r \) is the radius. Solving for \( r \), we get:\[ r = \frac{C}{2\pi} = \frac{7.00 \times 10^{-2}}{2\pi} \approx 0.0111\,\mathrm{~m} \]

Step 2: Calculate the Area of the Coil

The area \( A \) of the coil can be computed using the formula for the area of a circle:\[ A = \pi r^2 \]Substituting the value of \( r \):\[ A = \pi (0.0111)^2 \approx 3.87 \times 10^{-4} \mathrm{~m}^2 \]

Step 3: Apply the Torque Formula

The maximum torque \( \tau \) on a current loop in a magnetic field is given by the formula:\[ \tau = nIAB \sin \theta \]where \( n \) is the number of turns (1 turn), \( I \) is the current (4.30 A), \( A \) is the area calculated earlier, \( B \) is the magnetic field (2.50 T), and \( \theta \) is the angle between the field and the normal to the loop (90°, so \( \sin \theta = 1 \)). Thus:\[ \tau = 1\times 4.30 \times 3.87 \times 10^{-4} \times 2.50 \times 1 \]Calculating gives:\[ \tau \approx 4.17 \times 10^{-3} \mathrm{~Nm} \]

Step 4: State the Result

After substituting and calculating all the values, the maximum torque experienced by the coil is approximately \( 4.17 \times 10^{-3} \mathrm{~Nm} \).

Key Concepts

Circular Coil

A circular coil is essentially a loop of wire that forms a complete circle. In this exercise, the wire's length determines the coil's circumference. By wrapping the wire into one complete loop, you create the circular coil.
To find the radius of the coil, use the formula for circumference:

• The formula is: \( C = 2\pi r \), where \( C \) is the wire's length.
• Rearrange it to find the radius: \( r = \frac{C}{2\pi} \).
• Substitute the given length (\(7.00 \times 10^{-2} \; m\)) into the formula to find \( r \approx 0.0111 \; m \).

This radius is then used to calculate other properties of the coil, such as its area, which is crucial in further calculations.

Current Loop

The concept of a current loop is central to understanding how coils interact with magnetic fields. A current loop is simply a circular path through which electrical current flows. Here, the loop consists of just one turn of the wire.

• Current flowing through the wire generates a magnetic field around it.
• The flow of 4.30 A in this loop is what allows it to experience a force when placed in an external magnetic field.
• This force leads to the generation of torque, which depends on the loop's area and the amount of current.

Understanding current loops helps in visualizing how electrical circuits can interact with fields in practical applications.

Magnetic Field

A magnetic field is a force field created by moving electric charges, such as those in a current-carrying wire. In this case, the coil is placed in an external magnetic field of 2.50 T (tesla).

• Magnetic fields exert forces on other magnetic objects or moving charges within the field.
• The strength of this field influences how much torque the coil will experience.
• When fields interact with current loops, they can cause rotational motion, which is measured as torque.

Visualizing the magnetic field and its interaction with a current loop is key to understanding how motors and generators work.

Torque Calculation

Torque is the rotational effect produced by a force. For a current loop in a magnetic field, the torque can be calculated using the formula: \[ \tau = nIAB \sin \theta\]This formula tells us how the coil's dimensions and the magnetic field strength will affect the torque experienced.

• \( n = 1 \), since there's only a single loop.
• \( I = 4.30 \; A \), the current flowing through the coil.
• \( A = 3.87 \times 10^{-4} \; m^2 \), the area of the coil.
• \( B = 2.50 \; T \), the magnetic field strength.
• \( \sin \theta = 1 \) since the angle \( \theta = 90^\circ \), maximizing the perpendicular alignment.

Using these values, we calculated the maximum torque as approximately \( 4.17 \times 10^{-3} \; Nm \). This tells us how effectively the coil will rotate within the magnetic field.