Bar Magnets and Gauss's Law for Magnetism

The Bar Magnet

A bar magnet is one of the simplest and most familiar types of magnets. It is a rectangular piece of ferromagnetic material (like iron, nickel, cobalt, or their alloys) that has been permanently magnetised. It has two poles, conventionally called the North pole (N) and the South pole (S), located near its ends. The magnetic strength is concentrated at these poles.

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The Magnetic Field Lines

We can visualise the magnetic field produced by a bar magnet using magnetic field lines. By convention, magnetic field lines are considered to emerge from the North pole and enter the South pole outside the magnet. Inside the magnet, the field lines travel from the South pole to the North pole, forming continuous closed loops.

Magnetic field lines around a bar magnet form closed loops.

The properties of these magnetic field lines are consistent with the general properties of magnetic field lines discussed earlier (Section I1 of Magnetic Fields and Forces), including:

• Tangent to the field line gives the direction of $\vec{B}$.
• Density of field lines indicates field strength.
• Field lines never intersect.
• Field lines form closed loops.

The fact that magnetic field lines form closed loops is a fundamental property of magnetic fields, stemming from the absence of isolated magnetic poles (magnetic monopoles). Magnetic poles always exist in pairs (North and South). If you break a bar magnet into smaller pieces, each piece becomes a smaller magnet with its own North and South poles. You cannot isolate a single North or South pole.

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Bar Magnet As An Equivalent Solenoid

A long solenoid carrying a current produces a magnetic field that is strong and uniform inside, parallel to the axis, and weak outside, resembling the field of a bar magnet. In fact, the magnetic field pattern produced by a bar magnet is strikingly similar to the magnetic field pattern produced by a finite solenoid carrying a current.

Comparison of magnetic field lines for a bar magnet and a current-carrying solenoid.

This similarity is not just superficial. Theoretically, a bar magnet can be modelled as a collection of circulating atomic currents (due to electron motion) or as a collection of infinitesimal current loops distributed throughout its volume. Summing the magnetic fields produced by these microscopic loops results in the macroscopic magnetic field similar to that of a solenoid. Alternatively, one can imagine a bar magnet as being equivalent to a solenoid of the same dimensions carrying a suitable current. The end of the equivalent solenoid where current appears anticlockwise (when viewed from outside) acts as the North pole, and the end where it appears clockwise acts as the South pole.

Due to this equivalence, concepts like magnetic dipole moment, torque, and potential energy developed for current loops can be applied to bar magnets as well. A bar magnet can be considered as a magnetic dipole.

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The Dipole In A Uniform Magnetic Field

When a bar magnet (acting as a magnetic dipole with dipole moment $\vec{M}$) is placed in a uniform external magnetic field $\vec{B}$, it experiences a torque. The net force on a magnetic dipole in a uniform field is zero.

The torque on the magnetic dipole is given by:

$ \vec{\tau} = \vec{M} \times \vec{B} $

The magnitude of the torque is $\tau = MB \sin\theta$, where $\theta$ is the angle between the magnetic dipole moment vector $\vec{M}$ (which points from the South pole to the North pole of the magnet) and the magnetic field $\vec{B}$.

This torque tends to align the magnetic dipole moment $\vec{M}$ with the external magnetic field $\vec{B}$.

The potential energy of the magnetic dipole in the uniform magnetic field is given by:

$ U = -\vec{M} \cdot \vec{B} = -MB \cos\theta $

The potential energy is minimum when $\theta = 0^\circ$ (parallel alignment, stable equilibrium) and maximum when $\theta = 180^\circ$ (anti-parallel alignment, unstable equilibrium). A bar magnet in a uniform field will tend to orient itself to minimise its potential energy, aligning its North pole in the direction of the field.

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The Electrostatic Analog

Many concepts in magnetism involving magnetic dipoles in magnetic fields have direct analogies in electrostatics involving electric dipoles in electric fields. This analogy is very useful for understanding and formulating corresponding laws and principles.

Analogies between concepts in electrostatics and magnetism.

Electrostatics (due to charges)	Magnetism (due to currents/dipoles)
Electric charge ($q$)	Magnetic pole strength ($m$, hypothetical concept, or current element $I d\vec{l}$)
Electric field ($\vec{E}$)	Magnetic field ($\vec{B}$)
Electric dipole moment ($\vec{p} = q \vec{d}$)	Magnetic dipole moment ($\vec{M} = I \vec{A}$)
Coulomb's Law ($ \vec{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} $)	Force between magnetic poles (less fundamental, using pole strength) OR Biot-Savart Law ($ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3} $)
Electric force on a charge ($ \vec{F}_E = q\vec{E} $)	Magnetic force on a current element ($ d\vec{F}_B = I d\vec{l} \times \vec{B} $) OR on a moving charge ($ \vec{F}_B = q(\vec{v} \times \vec{B}) $)
Torque on an electric dipole ($ \vec{\tau} = \vec{p} \times \vec{E} $)	Torque on a magnetic dipole ($ \vec{\tau} = \vec{M} \times \vec{B} $)
Potential energy of an electric dipole ($ U_E = -\vec{p} \cdot \vec{E} $)	Potential energy of a magnetic dipole ($ U_B = -\vec{M} \cdot \vec{B} $)
Gauss's Law ($ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0} $)	Gauss's Law for Magnetism ($ \oint \vec{B} \cdot d\vec{A} = 0 $)
Permittivity of free space ($\epsilon_0$)	Permeability of free space ($\mu_0$)
Electric lines start/end on charges	Magnetic lines form closed loops (no magnetic monopoles)

While the analogy is powerful, it's important to remember that the fundamental source of magnetism is electric current (moving charges), not isolated magnetic monopoles like electric charges. Magnetic poles are just the manifestations of these underlying currents in magnetic materials. This is why magnetic field lines form closed loops and Gauss's Law for magnetism has a different form compared to Gauss's Law for electrostatics.

Magnetism And Gauss’S Law ($ \oint \vec{B} \cdot d\vec{A} = 0 $)

In electrostatics, Gauss's Law relates the electric flux through a closed surface to the net electric charge enclosed within that surface. It's a powerful statement about the nature of electric fields and the sources of electric fields (electric charges). There is a similar law for magnetic fields, known as Gauss's Law for Magnetism.

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Magnetic Flux ($ \Phi_B $)

Analogous to electric flux ($\Phi_E = \int \vec{E} \cdot d\vec{A}$), the magnetic flux ($\Phi_B$) through a surface is a measure of the number of magnetic field lines passing through that surface.

For a uniform magnetic field $\vec{B}$ passing through a flat surface of area $\vec{A}$ (where $\vec{A}$ is a vector perpendicular to the surface):

$ \Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta $

where $\theta$ is the angle between the magnetic field vector $\vec{B}$ and the area vector $\vec{A}$ (normal to the surface).

For a non-uniform magnetic field and an arbitrary surface, the magnetic flux through a surface S is given by the integral of the dot product of the magnetic field and the differential area vector over the surface:

$ \Phi_B = \int_S \vec{B} \cdot d\vec{A} $

The SI unit of magnetic flux is the weber (Wb). From the formula $\Phi_B = BA$, 1 Weber is equal to $1 \text{ Tesla} \times 1 \text{ meter}^2 = 1 \, T \cdot m^2$.

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Statement of Gauss's Law for Magnetism

Gauss's Law for Magnetism states that:

"The net magnetic flux through any closed surface is always zero."

Mathematically, this is expressed as:

$ \oint_S \vec{B} \cdot d\vec{A} = 0 $

Where the circle on the integral sign denotes a closed surface integral over any arbitrary closed surface $S$.

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Implication of Gauss's Law for Magnetism: Absence of Magnetic Monopoles

The most significant implication of Gauss's Law for Magnetism being zero is the fundamental principle that isolated magnetic poles (magnetic monopoles) do not exist.

In electrostatics, Gauss's Law ($\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{enclosed}}{\epsilon_0}$) is non-zero if there is a net electric charge enclosed within the closed surface. Electric field lines originate from positive charges and terminate on negative charges. Thus, if you enclose a positive charge with a closed surface, there will be a net outward electric flux.

In magnetism, magnetic field lines always form closed loops. For any closed surface, the number of magnetic field lines entering the surface must equal the number of magnetic field lines leaving the surface. There are no points where magnetic field lines begin or end. This is because there are no "magnetic charges" (magnetic monopoles) from which field lines originate or on which they terminate, analogous to electric charges. Magnetic dipoles are the simplest magnetic sources, and their field lines form closed loops. Even if you cut a magnet, you always get pairs of North and South poles, never an isolated pole.

Therefore, the net magnetic flux through any closed surface, regardless of what magnetic sources are inside (bar magnets, current loops, solenoids), will always be zero. Whatever flux enters the surface in some region must leave the surface in another region.

Gauss's Law for Magnetism is one of the four fundamental Maxwell's equations, which form the basis of classical electromagnetism.

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