Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consi

NCERT Exemplar · Q16

Q. Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consider the motion of

(i) electric dipole

p

in an electrostatic field

E

and

(ii) magnetic dipole

m

in a magnetic field

B

. Write down a set of conditions on

E

B

p

m

so that the two motions are verified to be identical. (Assume identical initial conditions.)

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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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The two motions match if the fields have the same spatial variation, the initial conditions are identical, and the magnitudes satisfy $pE = mB$ — then both the torque ( $\vec{p}\times\vec{E}=\vec{m}\times\vec{B}$ ) and the force ( $\nabla(\vec{p}\cdot\vec{E})=\nabla(\vec{m}\cdot\vec{B})$ ) are the same, so the trajectories coincide.

The two equations of motion

An electric dipole $\vec{p}$ in a field $\vec{E}$ and a magnetic dipole $\vec{m}$ in a field $\vec{B}$ obey structurally identical dynamics:

Quantity	Electric	Magnetic
Torque	$\vec{\tau} = \vec{p}\times\vec{E}$	$\vec{\tau} = \vec{m}\times\vec{B}$
Force	$\vec{F} = \nabla(\vec{p}\cdot\vec{E})$	$\vec{F} = \nabla(\vec{m}\cdot\vec{B})$
Energy	$U = -\vec{p}\cdot\vec{E}$	$U = -\vec{m}\cdot\vec{B}$

If the right-hand sides are equal at every point and instant, the equations of motion are the same; with identical initial conditions the trajectories then coincide.

Conditions for the forces and torques to coincide

1. Same field geometry. Choose $\vec{B}(\vec{r}) = \lambda\,\vec{E}(\vec{r})$ for a constant $\lambda$ over the region of motion — i.e. the two fields have the same spatial dependence (same direction and gradient everywhere), differing only by a constant scale.

2. Same orientation. Let $\vec{p}$ and $\vec{m}$ be oriented the same way relative to their fields, so the cross and dot products line up.

3. Magnitude matching. With $\vec{B}=\lambda\vec{E}$ , the torque condition $\vec{p}\times\vec{E} = \vec{m}\times\vec{B} = \lambda\,\vec{m}\times\vec{E}$ gives $p = \lambda m$ ; the force condition gives the same relation. Writing $\lambda = B/E$ ,

$p = \frac{B}{E}\,m \qquad\Longleftrightarrow\qquad pE = mB.$ …

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Method: Establishing Conditions for Two Analogous Dynamical Systems to Move Identically

Use this general technique whenever you're asked what conditions would make two structurally similar systems (here, an electric dipole in $\vec E$ and a magnetic dipole in $\vec B$ ) trace out exactly the same motion.

Steps

Step 1: Write the governing equations of both systems side by side

List every dynamical quantity (torque, force, potential energy) for each system in a table, using the analogous symbols:

Quantity	System 1	System 2
Torque	$\vec\tau=\vec p\times\vec E$	$\vec\tau=\vec m\times\vec B$
Force	$\vec F=\nabla(\vec p\cdot\vec E)$	$\vec F=\nabla(\vec m\cdot\vec B)$
Energy	$U=-\vec p\cdot\vec E$	$U=-\vec m\cdot\vec B$

Recognising that the two sets of equations have the same mathematical form is the whole basis of the analogy — Newton's second law (translational and rotational) will produce identical trajectories if, and only if, the right-hand sides are identical at every point and instant.

Step 2: Require the fields to have the same spatial profile

Set $\vec B(\vec r)=\lambda\,\vec E(\vec r)$ for some constant $\lambda$ , over the whole region of motion — i.e. the two fields must point the same way and vary with position in exactly the same pattern, differing only by an overall scale factor. Without this, the force/torque directions would differ at different points and the paths would diverge.

Step 3: Match the initial conditions

State explicitly that both systems must start from the same position, orientation, velocity, and angular velocity — otherwise even identical equations of motion produce different trajectories. …

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Q.Suppose we want to verify the analogy between electrostatic and magnetostatic by an explicit experiment. Consider the motion of

The two equations of motion

Conditions for the forces and torques to coincide

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