Question

A magnet of length $0.1\mathrm{m}$ and pole strength ${10}^{-4}\mathrm{A}\mathrm{m}.$ is kept in a magnetic field of 30 tesla at an angle of 300 . The Torque acting on it is $\times {10}^{-4}$ Joule. (a) $7.5$ (b) 3 (c) $1.5$ (d) 6

Short answer

The correct answer is (c) $1.5\ast {10}^{-4}$ Joule.

Step-by-step solution

Calculate magnetic moment

To calculate the magnetic moment (m), we use the formula: Magnetic moment (m) = pole strength (p) * length (l) where p = ${10}^{-4}$ A·m and l = 0.1 m. m = ${10}^{-4}A·m\ast 0.1m$ m = ${10}^{-5}A·m^2$

Calculate the torque

Now, we need to find the torque (τ) acting on the magnet using the formula: Torque (τ) = magnetic moment (m) * magnetic field (B) * sin(θ) where m = ${10}^{-5}A·m^2$, B = 30 T, and θ = 30°. First, we need to convert the angle to radians: θ (in radians) = (30° * π) / 180° θ = π/6 Now, we can calculate the torque: τ = ${10}^{-5}A·m^2\ast 30T\ast sin(\pi /6)$ τ = $3\ast {10}^{-4}N·m\ast 0.5$ τ = $1.5\ast {10}^{-4}N·m$

Compare the torque with the options

We have found that the torque acting on the magnet is $1.5\ast {10}^{-4}N·m$. Comparing this value with the given options: (a) 7.5 * ${10}^{-4}$ Joule (b) 3 * ${10}^{-4}$ Joule (c) 1.5 * ${10}^{-4}$ Joule (d) 6 * ${10}^{-4}$ Joule So the correct answer is (c) 1.5 * ${10}^{-4}$ Joule.

Key concepts

Magnetic Moment

The magnetic moment is a fundamental concept in magnetism that describes the strength and direction of a magnetic source. For a magnet, the magnetic moment is a measure of its ability to exert a torque in a given magnetic field. It's calculated using the formula:

1. Magnetic moment ($m$ ) = pole strength ($p$ ) × length ($l$ )

In this exercise, the pole strength is given as ${10}^{-4}\text{A}\cdot \text{m}$ and the length of the magnet is $0.1\text{m}$. The resulting magnetic moment becomes ${10}^{-5}\text{A}\cdot {\text{m}}^2$. The magnetic moment's unit, ampere meter squared (A·m²), captures both the strength (current component) and the spatial orientation (length) of the magnetic element.

The direction of the magnetic moment is from the south to the north pole of the magnet when visualized as a vector. This directional quality is crucial when calculating interactions in external magnetic fields, as it determines the force and torque experienced by the magnet.

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. In simpler terms, it's a field around a magnet where magnetic forces can be detected. Magnetic fields are typically measured in Tesla (T), a unit that quantifies their strength.

In the given problem, the magnetic field is 30 T. This is a relatively strong field where the magnet experiences forces depending on its orientation and magnetic moment. When a magnetic object is placed in an external magnetic field, the field induces a magnetic force on the object, which can lead to a torque (a rotational force) if the magnetic moment is not aligned with the field. This torque tends to align the magnetic moment parallel to the magnetic field.

Understanding magnetic fields can also help explain phenomena such as electric current generation in conductors (electromagnetic induction), the operation of motors and generators, and the principles behind technologies like MRI (Magnetic Resonance Imaging). The interplay between magnetic moments and magnetic fields is what drives these interactions.

Angle in Radians

Angles are crucial in trigonometry, often influencing calculations in physics involving periodic or wave phenomena, such as oscillations, circles, and rotations. Instead of degrees, angles are frequently measured in radians in scientific contexts. This choice is due to radians providing a more direct relationship with the arc length of a circle.

A complete circle is $2\pi$ radians, equivalent to 360 degrees. Thus, one radian is approximately 57.3 degrees. The exercise mentions converting $30$ degrees into radians to facilitate the calculation of torque. This conversion is performed using the expression:$$\theta =\left(\frac{\pi }{180}\right)\times 30\text{degrees}$$$$\theta =\frac{\pi }{6}\text{radians}$$The use of radians streamlines calculations, particularly when dealing with trigonometric functions like sine, cosine, or tangent, as evidenced in the torque formula $\tau =m\cdot B\cdot \sin (\theta )$. Here, $\sin$ refers to the sine function, with radians giving precise mathematical results essential for expressing the torque accurately when coupled with the magnetic moment and magnetic field. Understanding angles in radians is pivotal as it reaches broader applications across physics and engineering disciplines.