Dipole Uniform Magnetic Field

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In this article, you will get to 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 nearby 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 (B).

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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.

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.

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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\[^{→}\] ∗ B\[^{→}\] 

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

If θ = 900 and B = 1

Then we can obtain from equation (2),

 τ = M 

Thus, the torque, which is essential to keep the magnet at 900 with a magnetic field, is equal to the magnetic moment induction.

Electric Dipole in Uniform Magnetic Field

For a couple opposite charges having magnitude (q), the electric dipole moment can be explained as the product of the value of the charge with the distance between them, and the defined direction is towards the positive charge.

This is obvious from the information that a compass needle when rotating around these two bodies show similar deflection.

Afterward, noticing the close resemblance between these two, Ampere validated that a simple current loop acts like a bar magnet and set forth that all the magnetic singularities result from the circulation of the electric current. This is known as Ampere’s hypothesis.

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A current loop’s magnetic moment is defined as the multiplication of the current and the loop area. Its direction is perpendicular to the plane of the loop.

The dipole in a uniform magnetic field electrostatic analog

We accomplish that the magnetic field because of a bar magnet at an enormous distance is analogous to that of an electric dipole in an electric field. The analogic relation can be stated as given below,

 E →B, p →m, 1/4π∈0 → μ0 /4π 

If the value of r, which is the length of the point from the given magnet is very large as compared to the size of the magnet given by I, then we can rewrite the equatorial field generated by a bar magnet as,

BE = -μ0m/4πr3

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

BA = -μ02m/4πr3

FAQ (Frequently Asked Questions)

Q1. A Bar Magnet Having Moment 0.31 JT⁻¹ is Kept in a Uniform External Magnetic Field of 0.14 T. If the bar is Allowed to Rotate in the Plane of the Field, Which Coordination Would Agree to its (a) Stable (b) Unstable Equilibrium? Find the Potential Energy of the Magnet in Both Cases?

Ans: Given M = 0.31 JT⁻¹

B = 0.14 T

Θ = Angle between the bar magnet and the magnetic field

In stable equilibrium, Θ = 0⁰  

P.E= - M ∗ B Cos Θ = - 0.31∗ 0.14 ∗ Cos 0⁰ = - 4.3 ∗ 10⁻² J

In unstable equilibrium, Θ = 180⁰

P.E=- M ∗ B Cos 180 = - 0.31∗ 0.14 ∗ Cos 180⁰ = 4.3 ∗ 10⁻² J

Q2. Explain about EMF and Give its Formula?

Ans: EMF is known as Motional Electromotive Force. An EMF persuaded by the motion of the conductor through the magnetic field is a motional electromotive force.

The equation is given by E = -v ∗ L ∗ B.

Q3. Is Electric Dipole Moment Scalar or Vector?

Ans: It is a vector quantity. The charge direction is from the negative to the positive.

In vector form, it is written as: 

p = q ∗ d 

Q4. What is the Dipole Moment? Give an Example?

Ans: Polar molecules show a huge difference in the electrical charge at the positive end, as well as at the negative end. Alternatively, it is known as a dipole moment.

For illustration, ammonia (NH₃) is a polar molecule. This NH₃  contains one nitrogen atom covalently bonded to three hydrogen atoms as a result of the dipole moment.