A bar magnet of magnetic moment $m$ and moment of inertia $I$ (about centre, perpendicular to length) is cut into two equal pieces, perpendicular to length. Let $T$ be the period of oscillations of the original magnet about an axis through the mid point, perpendicular to length, in a magnetic field $B$. What would be the similar period $T'$ for each piece?

A bar magnet of magnetic moment m and moment of inertia I (about centre, perpendicular to length) is cut into

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Concept understanding — Magnetic Poles

Magnetic Poles: The Intuition First

Imagine you have a bar magnet — the kind you might have stuck on your refrigerator. If you bring two of them close, something interesting happens. Sometimes they snap together with a satisfying click. Other times, they push each other away, refusing to touch no matter how hard you try.

That's not random. Every magnet has two special regions, one at each end, where the magnetic force is strongest. These are its magnetic poles.

Note

The word "pole" comes from the Greek polos, meaning "pivot" or "axis" — the Earth itself has a North Pole and a South Pole, and it behaves like a giant magnet.

The Two Types of Poles

Every magnet has exactly two poles: a north pole and a south pole. You cannot have a magnet with only one pole — cut a bar magnet in half, and each half immediately becomes a complete magnet with its own north and south poles.

The rule of interaction is simple and memorable:

Unlike poles attract: north pulls south, south pulls north.
Like poles repel: north pushes north away; south pushes south away.

This is the fundamental behaviour. No exceptions.

The Precise Statement

Magnetic poles are the regions of a magnet where the external magnetic field is strongest. Every magnet has exactly two poles — a north pole and a south pole — that cannot be isolated. Like poles repel; unlike poles attract.

The key points to remember for exams:

Poles always come in pairs — there is no magnetic monopole (a single isolated pole) in nature, despite decades of searching.
The north pole is defined as the pole that points toward Earth's geographic north when the magnet is freely suspended.
The south pole points toward Earth's geographic south.

A Common Confusion (Watch Out)

Watch out

Earth's geographic North Pole is actually a magnetic south pole. Why? Because the north pole of a compass needle (which is a magnetic north pole) is attracted to it. And unlike poles attract. So the Earth's north pole behaves like a magnetic south pole. This often trips students up in exams.

Why This Matters …

Why this formula?

Magnetic Poles: Why the Key Formulas Hold

Let's build this from first principles — understanding why a magnetic pole behaves the way it does, not just memorizing the result.

1. What Is a Magnetic Pole?

A magnetic pole is a conceptual point where the magnetic field appears to originate or terminate. In reality, magnetic poles always come in north-south pairs (no isolated monopoles exist in nature), but we treat them as idealized sources for calculations.

North pole: source of magnetic field lines (outward)
South pole: sink of magnetic field lines (inward)

2. The Key Formula: Force Between Two Magnetic Poles

The force between two magnetic poles of strengths $m_1$ and $m_2$ , separated by distance $r$ , is:

F = \frac{\mu_0}{4\pi} \cdot \frac{m_1 m_2}{r^2}

Why this form?

This is a Coulomb's law analog — and that's not a coincidence. Here's the reasoning:

Experimental observation: Magnetic poles attract/repel with a force that:
- Varies as $1/r^2$ (inverse square law)
- Is proportional to the product of pole strengths
- Depends on the medium (via $\mu_0$ , the permeability of free space)
Mathematical analogy: The magnetic field $B$ at distance $r$ from a single pole $m$ is:

B = \frac{\mu_0}{4\pi} \cdot \frac{m}{r^2}

This comes from Gauss's law for magnetism applied to a point source.

Force derivation: The force on pole $m_2$ in the field of pole $m_1$ is:

F = m_2 \cdot B_1 = m_2 \cdot \left( \frac{\mu_0}{4\pi} \cdot \frac{m_1}{r^2} \right)

Hence:

F = \frac{\mu_0}{4\pi} \cdot \frac{m_1 m_2}{r^2}

Key insight: The $1/r^2$ dependence is not arbitrary — it follows from the geometry of 3D space (flux spreads over a sphere of area $4\pi r^2$ ).

3. The Magnetic Field of a Bar Magnet (Two Poles)

For a bar magnet of length $2l$ with poles $+m$ and $-m$ , the field at a point on the axis at distance $x$ from the center is:

B = \frac{\mu_0}{4\pi} \cdot \frac{2m l}{(x^2 - l^2)^2}

Why this form?

Superposition principle: The total field is the vector sum of fields from the north pole ( $+m$ ) and south pole ( $-m$ ).
Field from north pole at distance $(x - l)$ :

B_N = \frac{\mu_0}{4\pi} \cdot \frac{m}{(x - l)^2} \quad \text{(away from north)}

Field from south pole at distance $(x + l)$ :

B_S = \frac{\mu_0}{4\pi} \cdot \frac{m}{(x + l)^2} \quad \text{(toward south)}

Net field (both along same direction on axis):

B = B_N - B_S = \frac{\mu_0 m}{4\pi} \left[ \frac{1}{(x - l)^2} - \frac{1}{(x + l)^2} \right]

Simplify using algebra:

\frac{1}{(x - l)^2} - \frac{1}{(x + l)^2} = \frac{4xl}{(x^2 - l^2)^2}

Therefore:

B = \frac{\mu_0}{4\pi} \cdot \frac{4mxl}{(x^2 - l^2)^2}

But for a bar magnet, the magnetic moment is $\vec{M} = m \cdot (2l)$ (pole strength × separation). So $2ml = M$ , giving:

B = \frac{\mu_0}{4\pi} \cdot \frac{2Mx}{(x^2 - l^2)^2}

Key insight: The field is not simply $1/r^2$ because we have two poles — the net effect is a dipole field, which falls off as $1/r^3$ at large distances.

4. The Far-Field Approximation (Dipole Formula)

For $x \gg l$ (far from the magnet), $x^2 - l^2 \approx x^2$ , so:

B \approx \frac{\mu_0}{4\pi} \cdot \frac{2M}{x^3}

Why $1/x^3$ ?

A single pole gives $1/r^2$ …

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When a bar magnet is cut perpendicular to its length, each piece retains the same pole strength but has half the length, so the magnetic moment halves and the moment of inertia reduces by a factor of 8. The period of oscillation depends on $\sqrt{I/m}$ , giving $T' = T/2$ .

The key insight here is that magnetic moment and moment of inertia both change when you cut the magnet, and they change by different factors. The period of a magnetic dipole oscillating in a uniform field depends on the ratio $I/m$ , so we need to track how that ratio transforms.

Why this approach works

A bar magnet oscillating in a uniform magnetic field behaves like a torsional pendulum. The restoring torque is $\tau = -mB\sin\theta \approx -mB\theta$ for small angles, giving angular SHM with period:

T = 2\pi\sqrt{\frac{I}{mB}}

Since $B$ is the same for both the original and the pieces, the period ratio depends only on how $I$ and $m$ change when we cut the magnet.

Step-by-step reasoning

1. What changes when you cut perpendicular to length?

Cutting perpendicular to length means you slice the magnet into two equal halves along its short axis — like cutting a pencil into two shorter pencils. Each piece has:

Half the original length: $L' = L/2$
Same cross-sectional area (the cut doesn't change the width or thickness)
Same material, so same magnetization per unit volume

2. Magnetic moment halves

Magnetic moment $m$ of a bar magnet equals pole strength $p$ times length $L$ : $m = pL$ .

When you cut the magnet, each piece still has the same pole strength $p$ at its ends (pole strength depends on the cross-section and material, which haven't changed). But the length is halved, so:

m' = p \cdot \frac{L}{2} = \frac{m}{2}

Watch out

A common mistake is thinking pole strength also halves. It doesn't — pole strength is an intrinsic property of the magnet's end faces, which remain the same size and material after cutting.

3. Moment of inertia changes by a factor of 8

The original magnet has moment of inertia $I$ about its centre, perpendicular to length. For a uniform rod of mass $M$ and length $L$ :

I = \frac{1}{12}ML^2

Each piece has: …

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Concept understanding — Magnetic Poles

Magnetic Poles: The Intuition First

The Two Types of Poles

The Precise Statement

A Common Confusion (Watch Out)

Why This Matters …

Why this formula?

Magnetic Poles: Why the Key Formulas Hold

1. What Is a Magnetic Pole?

2. The Key Formula: Force Between Two Magnetic Poles

Why this form?

3. The Magnetic Field of a Bar Magnet (Two Poles)

Why this form?

4. The Far-Field Approximation (Dipole Formula)

Why $1/x^3$ ?

Why this approach works

Step-by-step reasoning

Method: Finding How the Oscillation Period of a Magnet Scales When It Is Cut or Resized

Steps

Why this approach works

Step-by-step reasoning

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