A sphere of radius \(a\) is magnetized uniformly parallel to a diameter so, that its magnetic moment per unit volume is \(M\). Find the B-field at a distance \(r\) from the centre for \(r>a\) and \(r

The magnetic moment is a fundamental concept in understanding magnetism and is often introduced when learning about how objects can influence magnetic fields. It is a vector quantity, represented by the symbol \( \mu \), that measures the strength and direction of a magnetic source.

In the context of magnetostatics, which is the study of magnetic fields in systems with steady currents or permanent magnets, the magnetic moment characterizes the entire magnetic behavior of a magnetized object. For example, when a sphere is uniformly magnetized, each unit volume contributes a small magnetic moment to the whole. These moments are directed along the magnetization direction and sum up to give the sphere's total magnetic moment.

To calculate the total magnetic moment, as mentioned in the step-by-step solution, one multiplies the magnetic moment per unit volume, \( M \), by the volume of the sphere. Mathematically, this is \( \mu = M \times \frac{4}{3}\pi a^3 \), where \( a \) is the radius of the sphere. Understanding this concept is crucial when dealing with the resulting magnetic field both inside and outside the magnetized object.

One interesting trait about magnetic moments is that they are the source of magnetic fields in the absence of electric currents. In our everyday life, this property is utilized in various devices such as compass needles, which align with the Earth's magnetic field due to their own magnetic moments.

Magnetostatics is a branch of physics focused on the behavior of magnetic fields in systems where the currents are steady, meaning they don't change with time. This field of study rests on the assumption that steady currents produce static magnetic fields, which is a reasonable approximation in many real-world scenarios.

The governing laws of magnetostatics include Ampere's Law and the magnetic field analogue of Gauss's Law for electricity, which states that the net magnetic flux through a closed surface is always zero. This is because magnetic fields do not originate or terminate at any point; instead, they form closed loops. A magnetized sphere, as in the textbook example, falls under the category of problems solvable with these laws since the magnetization produces a static and predictable framework for magnetic fields.

In the provided solution, magnetostatics is applied by using the dipole approximation to calculate the magnetic field inside and outside the sphere. This approach simplifies the problem, allowing for the use of expressions derived for static electric charges, which are closely related to their magnetic counterparts in terms of mathematical form.

Relevance in Everyday Life

Understanding magnetostatics isn't just for academic exploration; it's integral in designing electromagnetic devices, like electric motors and generators, which operate on the principles of steady magnetic fields.

In the study of magnetic fields, the dipole approximation is a simplifying assumption that allows for an easier computation of fields around magnetic sources. The approximation considers the magnetic source as a magnetic dipole with a north and a south pole, analogous to the positive and negative charges in electric fields.

This approximation is valid when the distance from the magnetic source is relatively large compared to the size of the source itself. Consequently, the complex distribution of the magnetic material is reduced to a simple model, characterized solely by its magnetic moment.

When applying the dipole approximation to our problem of a magnetized sphere, the sphere behaves like a bar magnet with a north and south pole at opposite ends of a diameter. This treatment is exceptionally useful because it reduces a potentially complicated three-dimensional magnetized volume to a point dipole located at the center for calculations outside the sphere (\(r > a\)).

Inside the sphere (\(r < a\)), the dipole approximation still holds, but with a different interpretation. The magnetic field at any point inside is due to the magnetization of the volume of the sphere enclosed by the radius \(r\). No need to consider the rest of the sphere material outside this radius because, as per the magnetic Gauss’s Law, the contribution cancels out.

By mastering the dipole approximation, students can tackle a wide variety of magnetic field problems with greater ease, a testament to the power of mathematical simplifications in physics.