Question

A loop of wire of radius \(R=25.0 \mathrm{~cm}\) has a smaller loop of radius \(r=0.900 \mathrm{~cm}\) at its center such that the planes of the two loops are perpendicular to each other. When a current of \(14.0 \mathrm{~A}\) is passed through both loops, the smaller loop experiences a torque due to the magnetic field produced by the larger loop. Determine this torque assuming that the smaller loop is sufficiently small so that the magnetic field due to the larger loop is same across the entire surface.

Short answer

Question: Calculate the torque experienced by a small loop of wire with a radius of 0.900 cm placed in the magnetic field produced by a larger loop of wire with a radius of 25 cm, when the planes of the loops are perpendicular to each other and the current is 14 A. Answer: The torque experienced by the smaller loop is approximately \(3.34 × 10^{-6}\) Nm.

Step-by-step solution

Calculation of the magnetic field at the center of the larger loop

We begin by calculating the magnetic field (B) at the center of the larger loop. The magnetic field produced by a current-carrying loop at its center is given by: \(B = \frac{\mu_{0} I}{2R}\) where \(μ_{0}\) is the permeability of free space, \(I\) is the current in the loop, and \(R\) is the radius of the loop. We can plug in the values to find the magnetic field: \(B = \frac{4π × 10^{-7} \text{ T m/A} × 14\text{ A}}{2 × 0.25\text{ m}}\)

Calculation of the torque on the smaller loop

The torque experienced by a current-carrying loop in a magnetic field is given by: \(τ = IABsinθ\) where \(τ\) is the torque, \(I\) is the current, \(A\) is the area of the loop, \(B\) is the magnetic field, and \(θ\) is the angle between the area vector and the magnetic field vector. Since the planes of the loops are perpendicular to each other, \(\theta = 90°\) so, \(τ = IABsin90° = IAB\) The area of the smaller loop is given by: \(A = πr^2\) Plugging in the values for the magnetic field and area, the torque can be calculated: \(τ = (14\text{ A})(π (0.009\mathrm{~m})^2)(\frac{4π × 10^{-7} \text{ T m/A} × 14\text{ A}}{2 × 0.25\text{ m}})\)

Calculate the result and simplify

After plugging in all values, we can calculate the torque: \(τ = 14\text{ A}(π (0.009\mathrm{~m})^2)(\frac{4π × 10^{-7} \text{ T m/A} × 14\text{ A}}{2 × 0.25\text{ m}})\) \(τ = 14π (0.000081\text{ m}^2)(\frac{28π × 10^{-7} \text{ T m}}{0.5\text{ m}})\) \(τ ≈ 3.34 × 10^{-6} \text{ N m}\) So the torque experienced by the smaller loop is \(3.34 × 10^{-6}\) Nm.

Key concepts

Magnetic Field at Loop Center

Understanding the magnetic field at the loop center is crucial when dealing with electromagnetic phenomena. In our scenario, the larger loop generates a magnetic field that affects the smaller loop situated at its center. According to Biot-Savart Law, a current-carrying loop creates a magnetic field through its center that's directly proportional to the current and inversely proportional to its radius.

The formula utilized for this calculation is:\[B = \frac{\mu_{0} I}{2R}\]where \(B\) is the magnetic field, \(\mu_{0}\) represents the permeability of free space, \(I\) is the electric current, and \(R\) is the radius of the larger loop.

The permeability of free space is a fundamental physical constant denoting the ability of a vacuum to support magnetic fields. It is precisely defined and has a value of \(4\pi \times 10^{-7} \text{ T m/A}\). Utilizing these concepts, when the current and radius are known, the magnetic field strength can be computed easily, and it facilitates the calculation of electromagnetic effects such as torque on a loop.

Torque on Current-Carrying Loop

Moving on to the torque exerted on a current-carrying loop, it's pivotal to recognize the factors contributing to this effect. Torque is the rotational force causing an object to spin around an axis. The torque \(\tau\) experienced by a current-carrying loop within a magnetic field is determined by the formula:\[\tau = IAB\sin(\theta)\]where \(I\) is the current, \(A\) is the area of the loop, \(B\) is the magnetic field at the loop's position, and \(\theta\) is the angle between the plane of the loop and the direction of the magnetic field.

Importance of Loop Orientation

In our exercise, the smaller loop's plane is perpendicular to the plane of the larger loop, which implies that the angle \(\theta\) is 90 degrees. Consequently, \(\sin(\theta)\) becomes 1, and the formula simplifies to \(\tau = IAB\). This simplification is crucial, as it means the torque is maximized when the loop is perpendicular to the magnetic field.

Permeability of Free Space

The permeability of free space \(\mu_{0}\) is a constant that appears frequently in equations dealing with electromagnetism. Its value, \(4\pi \times 10^{-7} \text{ T m/A}\), enables us to calculate the magnetic field strength in a vacuum. Importantly, it serves as a proportionality constant in the formula for the magnetic field at the loop's center, as well as in other magnetic field calculations.

Role in Electromagnetic Theory

It's integral to Maxwell's equations, which govern the behavior of electric and magnetic fields. The permeability of free space helps to quantify the relationship between electric current, magnetic fields, and the physical space where these interactions occur. In practical applications, this constant is used alongside the current and geometric factors of the loop to compute the resulting magnetic field, leading to further calculations of electromagnetic phenomena such as inductance, force, and, in our case, torque.

Angular Relation in Torque

The angular relationship in torque plays a significant role in determining the magnitude of the torque. This relationship is expressed in the torque formula through the sine function, revealing how the angle between the magnetic field and the area vector impacts the torque experienced by the loop.

For the smaller loop in our example, which is placed perpendicularly to the larger loop's plane, this angular relation contributes to the simplicity of the torque calculation. The angle, in this case, is \(90\) degrees, making \(\sin(\theta)\) equal to \(1\), and the torque equation reduces to the product of current, area, and magnetic field strength. Understanding this angular relation is key to grasping why changing the orientation of the loop with respect to the magnetic field would alter the torque it experiences.