Dipole Uniform Magnetic Field

In this article, you will learn the behavior of the forces performing on a dipole in a uniform magnetic field and will correlate it with the situation when a dipole is retained in an electrostatic field. For example, we experience that if we keep the iron fillings near a bar magnet upon a piece of paper and pound the sheet, the fillings assemble themselves to create a particular design. Here, the arrangement of iron filings signifies the magnetic field lines produced by the magnet. These lines generated due to the magnetic field provide us a fairly accurate clue of the magnetic field. On the other hand, most often, we are prescribed to govern the amount of magnetic field B precisely. We achieve this by employing a small compass needle of identified magnetic moment (m) and moment of inertia and let it oscillate in that particular magnetic field.

The region around a magnet within which another magnet can influence is called the Magnetic Field . A series of lines around a magnet is represented as a magnetic field. The magnetic field is one of the most important topics of physics.

Apart from the book, we also use a magnetic field in our daily life. Let us get familiar with the term and enjoy the study of Vedantu. Students are going to have fun when they know about the uses. Here are some uses like

Computer hard disk

• Television

• Radio

• Telephone

• Microwaves oven.

A pair of two equal and opposite forces having a different line of action gives rise to a turning effect known as torque along the axis, which is perpendicular to the plane of the force.

Students can describe the magnetic influence on moving electric charges; electric currents and magnetic materials are magnetic fields. It is a vector field. A moving charge present in a magnetic field experiences a force perpendicular to its velocity and the magnetic field.

A magnetic field permanent magnets pull on ferromagnetic materials such as iron and attract or repel other magnets. Also, a magnetic field that varies with location will exert a force on a range in respect to non-magnetic materials by affecting the motion of their outer atomic electrons. Those Magnetic fields surrounding magnetized materials are created by electric currents such as those used in electromagnets and electric fields may vary in time. 

The strength and direction of a magnetic field may sometimes vary with location; it is described by a function assigning a vector mathematically to each point of space, called a vector field.

Torque on a Magnetic Dipole in a Uniform Magnetic Field

Usually, a Magnetic dipole is a small magnet of atomic to subatomic sizes, similar to a flow of electric charge around a loop. Electrons rotating on their axes, electrons passing around atomic nuclei, and spinning positively charged atomic nuclei all are magnetic dipoles.

The addition of these effects may cancel so that a specified type of atom may not be a magnetic dipole. If they do not fully cancel, the atom is an everlasting magnetic dipole. Such dipoles are iron atoms. Millions of iron atoms locked with the same arrangement spontaneously creating a ferromagnetic domain also create a magnetic dipole.

Magnetic compass needles, and magnetic bars are examples of macroscopic magnetic dipoles.

Let’s take a magnet bar (N-S) having the length 2l and the pole strength m located in a uniform magnetic field of induction denoted as B by creating an angle θ with the field direction, as shown in the figure below. Because of this magnetic field denoted by B, the first force (m ∗ B) executes on the North Pole along the magnetic field direction, and another force (m ∗ B) executes on the South Pole along the opposite direction to the magnetic field. These two new forces are identical and inverse,

Therefore it establishes a couple.

Torque on a Magnetic Dipole in a Uniform Field

When a magnetic rod, (which can be taken as a magnetic dipole), is kept in a uniform magnetic field, the North Pole senses a force equal to the multiplication of the magnetic field intensity and the pole strength in the magnetic field direction.

Nonetheless, the South Pole senses a force, equal in magnitude but opposite in direction. Hence a torque exerts on the magnetic dipole because of which the magnet starts to rotate.

The torque is denoted as τ because the couple is:

τ = Force ∗ Perpendicular distance 

= F ∗ NA------(1)

We know, F = m ∗ B

So,  = mB ∗ 2l sin θ

 = MB sin θ------(2)

It can be written in the vector form

as  τ  = \[M^{\rightarrow}* B^{\rightarrow}\] 

We also know that the direction of τ is perpendicular to the plane and;

If θ = \[90^{0}\] and B = 1

Then we can obtain from equation (2), τ = M 

Thus, the torque, which is essential to keep the magnet at \[90^{0}\] with a magnetic field, is equal to the magnetic moment induction.

Electrostatic Analog

Let's compare the equation of electric dipole in an electric field. We conclude that the magnetic field due to a bar magnet at a considerable distance is analogous to an electric dipole in an electric field. Likewise, the relation can be status as given below,

E→B,p→m,\[\frac{1}{4\pi\varepsilon_{0}}\rightarrow \mu_{0}4\pi E\rightarrow B\], ,p→m,\[\frac{1}{4\pi\varepsilon_{0}}\rightarrow \mu_{0}4\pi\]

If the value of r, that is, the distance of the point from the given magnet, is tremendous as compared to the size of the magnet given by I, or r >> l, then students can write the equatorial field generated by a bar magnet as,

B_E=−\[\frac{\mu_{0}m}{4\pi r}\]

3B_E=−\[\frac{\mu_{0}m}{4\pi r^{3}}\]

Comparably, the axial field of the bar magnet in the same condition can be given as,

B_A=−\[\frac{\mu_{0}2m}{4\pi r}\]

3B_A=−\[\frac{\mu_{0}2m}{4\pi r^{3}}\]

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