How Does a Dipole Behave in a 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,
BE=−\[\frac{\mu_{0}m}{4\pi r}\]
3BE=−\[\frac{\mu_{0}m}{4\pi r^{3}}\]
Comparably, the axial field of the bar magnet in the same condition can be given as,
BA=−\[\frac{\mu_{0}2m}{4\pi r}\]
3BA=−\[\frac{\mu_{0}2m}{4\pi r^{3}}\]
Stay updated with Vedantu to learn more about dipoles in a uniform magnetic field, the electrostatic analog, and other related topics.
FAQs on Understanding Dipole in a Uniform Magnetic Field
1. What is a uniform magnetic field and how is it typically represented?
A uniform magnetic field is a region in space where the magnetic field has the same strength and direction at every point. Visually, it is represented by a set of parallel, straight, and equally spaced magnetic field lines. This indicates that a magnetic particle would experience the same force regardless of its position within this field.
2. What happens when a magnetic dipole is placed in a uniform magnetic field?
When a magnetic dipole, such as a bar magnet or a current loop, is placed in a uniform magnetic field, it experiences a torque that tries to align its magnetic moment with the direction of the external field. Importantly, it experiences no net translational force because the force on the north pole is equal in magnitude and opposite in direction to the force on the south pole, causing them to cancel each other out.
3. What is the formula for the torque experienced by a magnetic dipole in a uniform magnetic field?
The formula for the magnitude of the torque (τ) on a magnetic dipole is given by: τ = mB sin(θ). In its vector form, it is expressed as a cross product: τ⃗ = m⃗ × B⃗. Here, 'm' is the magnetic dipole moment, 'B' is the magnitude of the magnetic field, and 'θ' is the angle between the magnetic moment vector and the magnetic field vector.
4. How is the potential energy of a magnetic dipole in a uniform magnetic field calculated?
The potential energy (U) of a magnetic dipole in a uniform magnetic field depends on its orientation relative to the field. The formula is given by: U = -mB cos(θ), which is the scalar (dot) product of the magnetic moment and the magnetic field vectors: U = -m⃗ ⋅ B⃗. The energy is lowest (most stable) when the dipole is aligned with the field (θ=0°) and highest (most unstable) when it is anti-aligned (θ=180°).
5. What is the difference between stable and unstable equilibrium for a magnetic dipole in a uniform field?
The equilibrium positions are where the net torque on the dipole is zero. There are two such positions:
- Stable Equilibrium: Occurs when the magnetic moment vector (m) is aligned with the magnetic field vector (B), meaning θ = 0°. At this point, the potential energy is at its minimum (-mB). If slightly displaced, the dipole will oscillate back to this position.
- Unstable Equilibrium: Occurs when the magnetic moment vector (m) is aligned opposite to the magnetic field vector (B), meaning θ = 180°. Here, the potential energy is at its maximum (+mB). Any slight displacement will cause the dipole to flip and align with the field.
6. Why does a magnetic dipole experience only a torque but no net force in a uniform magnetic field?
In a uniform field, the magnetic field strength (B) is constant everywhere. The north pole experiences a force (mB) in the direction of the field, while the south pole experiences a force of the same magnitude (mB) but in the exact opposite direction. These two equal and opposite forces form a couple, which creates a turning effect (torque) without any net linear push or pull. Therefore, the dipole rotates but does not accelerate linearly.
7. How does the behaviour of a magnetic dipole change if the magnetic field is non-uniform?
In a non-uniform magnetic field, the field strength varies from point to point. Consequently, the forces on the north and south poles are unequal in magnitude. This results in the dipole experiencing both a net torque (causing rotation) and a net force (causing it to move linearly towards the region of the stronger field). This is why a magnet can attract an unmagnetized object like an iron nail.
8. What are some real-world examples of a magnetic dipole in a uniform magnetic field?
Common examples illustrating this principle include:
- A magnetic compass needle aligning itself with the Earth's magnetic field (which is nearly uniform over small distances).
- The fundamental working principle of an electric motor, where a current-carrying coil (acting as a magnetic dipole) is placed in a magnetic field and experiences a torque that causes it to rotate, converting electrical energy into mechanical energy.
