Question

An iron rod of length \(L\) and magnetic moment \(\mathrm{M}\) is bent in the form of a semicircle. Now its magnetic moment will be (a) \(\mathrm{M}\) (b) \((2 \mathrm{M} / \pi)\) (c) \((\mathrm{M} / \pi)\) (d) \(\mathrm{M} \pi\)

Short answer

The magnetic moment of an iron rod of length L and magnetic moment M remains the same (M) when it is bent in the form of a semicircle because the magnetic moment is an intrinsic property of the material and does not change when the object changes shape. Therefore, the correct answer is (a) M.

Step-by-step solution

Understand the concept of magnetic moment

Magnetic moment is a measure of an object's ability to produce a magnetic field. We are given the magnetic moment of an iron rod (M) when it is straight. Now, we need to find how the magnetic moment changes when the rod is bent into a semicircle.

Consider the length of the rod

When the rod is bent into a semicircular shape, the length (L) becomes the circumference of the semicircle. The length of the rod (L) can be related to the radius (r) of the semicircle by using the formula: \[L = \pi r\] Now, we need to analyze how the magnetic moment changes with the shape.

Understand the effect of bending on magnetic moment

When an iron rod having magnetic moment M is bent, its magnetic moment remains constant because the magnetic moment is a property intrinsic to the material, and it doesn't change when the object changes shape. Thus, we can say that the magnetic moment of the rod (M) remains the same when it is bent into a semicircular shape.

Choose the correct answer

Given that the magnetic moment remains constant when the iron rod is bent, the correct option is: (a) M So, the magnetic moment of the iron rod bent into a semicircular shape is still M.

Key concepts

Iron Rod

An iron rod is a solid piece of iron that is often used in physics problems to study magnetic properties. Iron is a ferromagnetic material, which means it can easily be magnetized. This intrinsic property makes it crucial in creating strong magnetic fields. When an iron rod is magnetized, tiny regions called domains within the iron align, enhancing the overall magnetic field.

• Iron is ferromagnetic, meaning it can become strongly magnetized.
• The magnetic moment of iron is due to its atomic structure, where unpaired electrons contribute to the overall magnetic properties.

Understanding how an iron rod behaves in magnetic fields is fundamental for many technological applications, from simple magnets to complex electrical devices.

Semicircular Shape

A semicircular shape is formed when an object is bent to resemble half of a circle. In the context of a magnetic problem, bending an iron rod into a semicircular shape can change its spatial configuration but not its material properties. The length of the rod when bent forms the circumference of a semicircle, which can be related to its radius. This relationship is expressed mathematically as:
\[ L = \pi r \]
Where:

• \(L\) is the length of the iron rod
• \(r\) is the radius of the semicircle

Bending a rod changes how it is positioned in space but doesn’t affect its inherent magnetic properties.

Intrinsic Property

The intrinsic property refers to characteristics that are inherent and do not change with shape or size modifications. For an iron rod, its magnetic moment is an intrinsic property. The magnetic moment remains constant regardless of shape alteration. This principle implies that even when an object is bent or shaped differently, the magnetic moment, which is a measure of its ability to produce a magnetic field, does not change.

• The magnetic moment of the iron rod before and after bending is the same.
• Intrinsic properties are a key concept in understanding material behavior.

Recognizing intrinsic properties helps in solving physics problems by identifying what remains unchanged and focusing on those factors.

Magnetic Field Production

Magnetic field production is a fundamental concept in physics, detailing how magnetic moments contribute to creating magnetic fields. An iron rod with a magnetic moment is capable of generating a magnetic field around it. This field is visualized as lines of force that emanate from its poles. The strength and orientation of the magnetic field are determined by the magnetic moment.
Important points to consider include:

• The magnitude of the magnetic moment influences the strength of the magnetic field.
• The configuration of the field lines changes with the shape of the object producing them, although the moment itself remains constant.

Understanding magnetic field production is crucial for applications involving electromagnetism and is the basis for technologies like motors and transformers.