Verify the Gauss's law for magnetic field of a point dipole of dipole moment m at the origin for the surface w

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Concept understanding — Magnetic Field Lines

Magnetic Field Lines

A magnet or a current-carrying wire fills the space around it with a magnetic field. We cannot see this field, so we picture it using magnetic field lines — continuous curves that map both the direction and the strength of the field at every point.

What a field line represents

The tangent to a field line at any point gives the direction of the magnetic field $\vec{B}$ there. If you place a tiny compass needle at that point, it aligns along the tangent, its north pole pointing the way the line runs. The density of the lines (how closely packed they are) represents the magnitude of $\vec{B}$ : crowded lines mean a strong field, widely spaced lines mean a weak field.

Key properties (exam essentials)

Important

Outside a magnet the lines run from the north pole to the south pole, but they are continuous closed loops — inside the magnet they run south to north, so every line closes on itself.
Two field lines never intersect. If they did, a compass at the crossing point would have to point in two directions at once, which is impossible.
Lines are crowded where the field is strong (near the poles) and spread out where it is weak.
They form smooth, continuous curves with no free ends.

The fact that magnetic field lines always close on themselves is deep: it means there are no isolated magnetic poles (monopoles). This is Gauss's law for magnetism:

\oint \vec{B} \cdot d\vec{A} = 0

The net magnetic flux through any closed surface is zero — every line that enters the surface also leaves it.

Contrast with electric field lines

Note

Electric field lines start on positive charges and end on negative charges — they are open curves. Magnetic field lines have no such start or end; they are always closed loops. This single difference reflects that isolated electric charges exist, but isolated magnetic poles do not.

Uniform field …

Why this formula?

Magnetic Field Lines

A magnetic field line is an imaginary curve we draw to picture an invisible field. Its purpose is to encode two things at once: the direction of the field $\vec{B}$ (the tangent to the line at any point) and the strength of the field (how densely the lines are packed). They are a map, not physical objects.

The four defining rules

1. The tangent gives the field direction. At every point, $\vec{B}$ points along the tangent to the field line through that point. A compass needle placed on the line aligns with it.

2. Field lines form closed loops. Unlike electric field lines, which begin and end on charges, magnetic field lines never start or stop. This is Gauss's law for magnetism:

\oint \vec{B} \cdot d\vec{A} = 0

The net flux through any closed surface is zero because isolated magnetic poles (monopoles) do not exist — every north pole is paired with a south pole. So for a bar magnet the lines emerge from the north pole outside, curve around to the south pole, and continue through the interior of the magnet back to the north, closing the loop.

3. Field lines never cross. If two lines crossed, the tangent — and hence $\vec{B}$ — would have two directions at that point. Since the field has a single, unique direction everywhere, crossings are impossible. …

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The magnetic field of a point dipole obeys Gauss's law for magnetism ( $\nabla \cdot \mathbf{B} = 0$ ), so the total flux through any closed surface — including a sphere of radius $R$ — is exactly zero. The flux is $\boxed{0}$ .

Why This Works: The Magnetic Force Balance

Gauss's law for magnetism is not a mathematical coincidence — it's a physical statement that magnetic monopoles do not exist. For any magnetic field configuration, the net flux through a closed surface is always zero. This is fundamentally different from electric fields, where a point charge inside a surface gives non-zero flux.

For a point dipole at the origin, the field lines form closed loops: they emerge from the north pole, curve around, and re-enter at the south pole. Every field line that leaves the sphere must re-enter it somewhere else. The flux contributions from the outward and inward parts exactly cancel.

Step-by-Step Calculation

1. Write the magnetic field of a point dipole

The magnetic field of a point dipole $\mathbf{m}$ at the origin is:

\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \left[ \frac{3(\mathbf{m} \cdot \hat{\mathbf{r}})\hat{\mathbf{r}} - \mathbf{m}}{r^3} \right]

For a sphere of radius $R$ , the surface is at $r = R$ , and the outward normal is $\hat{\mathbf{n}} = \hat{\mathbf{r}}$ .

2. Set up the flux integral

The magnetic flux through the spherical surface is:

\Phi_B = \oint_S \mathbf{B} \cdot d\mathbf{a} = \oint_S \mathbf{B} \cdot \hat{\mathbf{r}} \, R^2 \, d\Omega

where $d\Omega = \sin\theta \, d\theta \, d\phi$ is the solid angle element.

3. Compute the radial component of $\mathbf{B}$

Take the dot product $\mathbf{B} \cdot \hat{\mathbf{r}}$ :

\mathbf{B} \cdot \hat{\mathbf{r}} = \frac{\mu_0}{4\pi R^3} \left[ 3(\mathbf{m} \cdot \hat{\mathbf{r}})(\hat{\mathbf{r}} \cdot \hat{\mathbf{r}}) - \mathbf{m} \cdot \hat{\mathbf{r}} \right]

Since $\hat{\mathbf{r}} \cdot \hat{\mathbf{r}} = 1$ , this simplifies to:

\mathbf{B} \cdot \hat{\mathbf{r}} = \frac{\mu_0}{4\pi R^3} \left[ 3(\mathbf{m} \cdot \hat{\mathbf{r}}) - (\mathbf{m} \cdot \hat{\mathbf{r}}) \right] = \frac{\mu_0}{4\pi R^3} \cdot 2(\mathbf{m} \cdot \hat{\mathbf{r}})

So:

\mathbf{B} \cdot \hat{\mathbf{r}} = \frac{\mu_0}{2\pi R^3} (\mathbf{m} \cdot \hat{\mathbf{r}})

4. Write the flux integral explicitly

\Phi_B = \oint \frac{\mu_0}{2\pi R^3} (\mathbf{m} \cdot \hat{\mathbf{r}}) \, R^2 \, d\Omega = \frac{\mu_0}{2\pi R} \oint (\mathbf{m} \cdot \hat{\mathbf{r}}) \, d\Omega

Tip

The factor $R^3$ in the denominator and $R^2$ from the area element combine to give $1/R$ , which will cancel with nothing — but the integral itself will vanish.

5. Evaluate the angular integral …

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Method: Verifying Gauss's Law for Magnetism ( $\oint\vec{B}\cdot d\vec{A}=0$ ) for a Given Field and Surface

Use this general technique whenever you're asked to explicitly verify (not just cite) that a magnetic field configuration satisfies $\oint\vec B\cdot d\vec A=0$ over some specified closed surface.

Steps

Step 1: Write the field in a coordinate system matched to the surface

If the closed surface is a sphere, express $\vec B$ in spherical coordinates centred on the source, since the outward normal of a sphere is simply $\hat r$ at every point. For a point dipole of moment $\vec m$ this is

\vec B(\vec r)=\frac{\mu_0}{4\pi}\left[\frac{3(\vec m\cdot\hat r)\hat r-\vec m}{r^3}\right].

Step 2: Write the area element in the same coordinates

For a sphere of radius $R$ : $d\vec A=\hat r\,R^2\,d\Omega$ , where $d\Omega=\sin\theta\,d\theta\,d\phi$ is the solid-angle element. This choice is deliberate — it makes $\vec B\cdot d\vec A$ collapse to just the radial component of $\vec B$ times $d\Omega$ , since every other component of $\vec B$ is automatically perpendicular to $d\vec A$ and drops out.

Step 3: Simplify $\vec B\cdot\hat r$ using $\hat r\cdot\hat r=1$

\vec B\cdot\hat r=\frac{\mu_0}{4\pi r^3}\left[3(\vec m\cdot\hat r)-(\vec m\cdot\hat r)\right]=\frac{\mu_0}{2\pi r^3}(\vec m\cdot\hat r).

This is the standard trick for any $\propto[3(\vec m\cdot\hat r)\hat r-\vec m]$ -type field: dotting with $\hat r$ always leaves a factor of $2(\vec m\cdot\hat r)$ .

Step 4: Integrate over the full solid angle using symmetry, not brute force …

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Q.Verify the Gauss's law for magnetic field of a point dipole of dipole moment mmm at the origin for the surface which is a sphere of radius RRR.

Why This Works: The Magnetic Force Balance

Step-by-Step Calculation

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Q.Verify the Gauss's law for magnetic field of a point dipole of dipole moment $m$ at the origin for the surface which is a sphere of radius $R$ .

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