Question

A small bar magnet of moment \(\mathrm{M}\) is placed in a uniform field of \(\mathrm{H}\). If magnet makes an angle of \(30^{\circ}\) with field, the torque acting on the magnet is (a) \(\mathrm{MH}\) (b) \((\mathrm{MH} / 2)\) (c) \((\mathrm{MH} / 3)\) (d) \((\mathrm{MH} / 4)\)

Short answer

The torque acting on the magnet is \(\frac{MH}{2}\).

Step-by-step solution

Identify the given values.

The information given is: - Magnetic moment, M - Magnetic field, H - Angle between the magnet and the field, θ = 30°

Apply the formula for torque.

The formula for torque acting on the magnetic dipole in a magnetic field is: τ = MH sin(θ) Plug in the given values: τ = M * H * sin(30°)

Calculate the sin value.

To find the torque, we first need to calculate the value of sin(30°). Using the sine function: sin(30°) = 1 / 2

Calculate the torque.

Now, substitute the sin value back into the equation: τ = M * H * (1 / 2) Simplify the equation: τ = (MH) / 2

Compare the result with the given options.

We have found that the torque acting on the magnet is (MH) / 2. This matches the option (b). Therefore, the correct answer is: (b) (MH / 2)

Key concepts

Magnetic Moment

The magnetic moment is a measure of the strength of a magnet and its ability to produce magnetic forces. It's an intrinsic property of the magnet, much like a compass needle has its own magnetic orientation. The larger the magnetic moment, the stronger the magnetic force that the magnet can exert. This term is essential in the interaction between magnets and external magnetic fields because it determines how much torque, or rotational force, the magnet experiences when placed in a magnetic field.

The magnetic moment is denoted by the symbol \( M \). It quantifies the magnet's overall magnetic resourcefulness. Essentially, it acts somewhat like the momentum of an object in a rotational sense. When you consider a compass needle flipping back and forth, you're seeing the magnetic moment in action.

Understanding the magnetic moment helps in solving problems involving magnetic torque, as it is directly proportional to the torque experienced by a dipole in a magnetic field.

Uniform Magnetic Field

A uniform magnetic field is a field with consistent magnitude and direction. Imagine a classroom fan blowing air in one direction at a constant speed. That's a bit like how a uniform magnetic field behaves with magnetic forces.

In a uniform magnetic field, every point of the field has the same strength and orientation. This makes calculations easier because you don't need to worry about changes in the field's strength or direction, simplifying the math.

When a magnetic dipole, such as a bar magnet, is placed in a uniform magnetic field, it interacts with the field uniformly. This means forces acting on the dipole due to the field are predictable and can be accurately calculated. This uniformity is why this setup is favored in many practical applications and theoretical studies involving magnetic interactions.

Sine Function

The sine function is a crucial part of trigonometry, often used in physics to resolve angles and forces. In the context of this problem, it helps determine the rotational force on a magnet placed in a magnetic field.

The sine function is denoted as \( \sin(\theta) \), where \( \theta \) is the angle between the magnetic moment and the magnetic field. It's a measure of how much of the magnetic moment is "inclined" towards the magnetic field direction, which directly impacts the torque produced.

For this exercise, calculating \( \sin(30°) \) is pivotal since it determines part of the torque's strength. The sine of 30 degrees is \( \frac{1}{2} \), meaning half of the magnetic moment contributes to the torque in the given direction. This is why the torque equation \( \tau = MH \sin(30°) \) simplifies to \( \tau = \frac{MH}{2} \). Understanding the sine function and its role in trigonometry is vital in physics for resolving vector components that often appear in physics problems.

Magnetic Dipole

A magnetic dipole is a magnetic entity that has two poles – a north and a south – and it's the simplest form of a magnet, much like a bar magnet. It represents the smallest and fundamental form of magnetism.

Dipoles create magnetic fields around them and are characterized by their magnetic moments. When placed in an external magnetic field, magnetic dipoles experience torque, which makes them rotate or align with the field. The interaction of magnetic dipoles with an external magnetic field is exactly what this exercise explores.

The concept of a magnetic dipole is essential because it simplifies the study of magnets and magnetic fields. Instead of dealing with complex systems of magnets, you can consider their effects using basic dipoles and analyze their behavior in uniform magnetic fields conveniently.

Overall, the notion of a magnetic dipole, its interaction with fields, and the concept of torque explain how magnetic moments manifest in rotational forces and contribute to the understanding of magnetism in practical scenarios.