Problem 38 - Physics

Question

Two coils have the same number of circular turns and carry the same current. Each rotates in a magnetic field as in Figure \(21-21\). Coil 1 has a radius of \(5.0 \mathrm{~cm}\) and rotates in a 0.18 -T field. Coil 2 rotates in a 0.42 - T field. Each coil experiences the same maximum torque. What is the radius (in \(\mathrm{cm}\) ) of coil \(2 ?\)

Step-by-Step Solution

Answer

The radius of coil 2 is approximately 3.27 cm.

Step 1: Understanding Torque in a Magnetic Field

The torque \( \tau \) on a coil in a magnetic field depends on the number of turns \( N \), the current \( I \), the magnetic field \( B \), and the area \( A \) of the coil. The equation for torque is given by:\[ \tau = N \cdot I \cdot B \cdot A \cdot \sin \theta \]. Where \(\theta\) is the angle between the normal to the coil and the magnetic field direction. For maximum torque, \(\sin \theta = 1\) and thus \[ \tau = N \cdot I \cdot B \cdot A \].

Step 2: Express Areas in Terms of Radius

The area \( A \) of a coil with radius \( r \) is \( \pi r^2 \). Thus, for each coil, the torque can be expressed as: \[ \tau_1 = N \cdot I \cdot B_1 \cdot \pi r_1^2 \] for coil 1, and \[ \tau_2 = N \cdot I \cdot B_2 \cdot \pi r_2^2 \] for coil 2.

Step 3: Equating Torques of Both Coils

Since both coils experience the same maximum torque, we can set \( \tau_1 \) equal to \( \tau_2 \):\[ N \cdot I \cdot B_1 \cdot \pi r_1^2 = N \cdot I \cdot B_2 \cdot \pi r_2^2 \]. Simplifying, we get: \[ B_1 \cdot r_1^2 = B_2 \cdot r_2^2 \].

Step 4: Solve for the Radius of Coil 2

Rearrange the equation from Step 3 to solve for \( r_2 \):\[ r_2^2 = \frac{B_1}{B_2} \cdot r_1^2 \]. Thus,\[ r_2 = \sqrt{\frac{B_1}{B_2}} \cdot r_1 \].

Step 5: Substitute Known Values

Substitute the given values: \( r_1 = 5.0 \) cm, \( B_1 = 0.18 \) T, \( B_2 = 0.42 \) T. Thus,\[ r_2 = \sqrt{\frac{0.18}{0.42}} \times 5.0 \].

Step 6: Calculate the Radius of Coil 2

Calculate the value using the formula from Step 5:\[ r_2 = \sqrt{\frac{0.18}{0.42}} \times 5.0 \approx \sqrt{0.4286} \times 5.0 \approx 0.6546 \times 5.0 \approx 3.27 \text{ cm} \].

Key Concepts

Magnetic Field

A magnetic field is an invisible force field that surrounds magnetic materials and moving electric charges. This field exerts a force on other nearby magnetic materials and moving charges. In the context of this exercise, the magnetic field is denoted as either \( B_1 \) or \( B_2 \) depending on the coil in question. Understanding how this field interacts with the coil is crucial for determining the torque experienced.

The strength of the magnetic field is measured in Tesla \( (T) \). A stronger magnetic field (higher value in Tesla) will exert a greater force on the coil, thus affecting the torque it experiences. For instance, Coil 1 is in a 0.18 T field, while Coil 2 is in a considerably stronger 0.42 T field.

### The Role of Magnetic Field in TorqueWhen a coil is placed in a magnetic field and carries current, the magnetic field interacts with the electric current to produce torque. Torque is the rotational force that causes the coil to rotate. The greater the magnetic field, the higher the potential torque, assuming other factors like current and coil area remain constant.

Current

The current, represented as \( I \) in equations, plays a significant role in generating torque when a coil is situated within a magnetic field. Current is essentially the flow of electric charge, and its presence in the coil generates a magnetic moment that allows the coil to interact with the external magnetic field.

### How Current Impacts TorqueThe torque \( \tau \) experienced by a coil is directly proportional to the current flowing through it, as shown in the torque equation \( \tau = N \cdot I \cdot B \cdot A \). Increasing the current will result in a greater torque for a coil within a magnetic field.

In this exercise, both coils have the same value of current, which means any differences in torque experienced by the coils are solely due to differences in other factors like the magnetic field strength \( B \) and the coil radius \( r \). Hence, current is essential but is held constant when comparing the two coils.

Coil Radius

The radius of a coil impacts the area it encompasses, which in turn affects the torque produced when the coil rotates in a magnetic field. The area of a coil is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius of the coil. Larger radii result in larger areas, leading to potentially greater torque if all other variables remain constant.

### Calculating and Comparing RadiiIn the given problem, the goal is to determine the radius \( r_2 \) of Coil 2. We know Coil 1's radius is 5.0 cm. By using the relationship \( r_2 = \sqrt{\frac{B_1}{B_2}} \cdot r_1 \), we can find \( r_2 \) by substituting the known values into the formula. Clearly, the radius is pivotal in calculating the torque and understanding how coils behave in magnetic fields.

Through these calculations, it becomes evident that radius adjustments are necessary when altering the magnetic field strength to maintain consistent torque across different coils.

Circular Turns

Circular turns in coils refer to the number of loops or rings the wire makes. More turns mean a larger total conductive length interacting with the magnetic field, enhancing torque. This feature is given by \( N \), the number of turns in the torque equation \( \tau = N \cdot I \cdot B \cdot A \). In the given exercise, both coils have the same number of turns, thus simplifying the torque comparison to other factors like magnetic field and radius.

### Importance of Circular TurnsWhile the number of turns is the same for both coils in this exercise, generally speaking, increasing the turns can amplify the torque without increasing the current or the field strength. However, too many turns might make the coil bulky and potentially increase resistance, which could limit the flow of current.

Understanding circular turns' role helps us appreciate their impact on magnetic interactions and design considerations for electromagnetic devices. Though equal turns do not affect the comparative torque in this exercise, they are a crucial variable in other practical scenarios involving magnetic coils.