# Magnetostatics in Vacuum

Physics II · Electromagnetism · https://tryals.app/learn/physics-ii/magnetostatics-in-vacuum

## A Field With No Sources

Magnetostatics mirrors electrostatics with the two roles swapped. Where $\mathbf{E}$ has divergence but no curl, $\mathbf{B}$ has curl but **no divergence**:

$$\nabla \cdot \mathbf{B} = 0, \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J}$$

with $\mu_0 = 4\pi \times 10^{-7}$ T m/A. The first says there are no magnetic monopoles: every field line closes on itself, so the flux through any closed surface is exactly zero. The second is **Ampère's law**, whose integral form is

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$

Because $\mathbf{B}$ is divergence-free it cannot be the gradient of a scalar, but it *can* be the curl of something, the **vector potential** $\mathbf{A}$, with $\mathbf{B} = \nabla \times \mathbf{A}$. In the Coulomb gauge this satisfies $\nabla^2\mathbf{A} = -\mu_0\mathbf{J}$, one Poisson equation per component.

Ampère's law earns its keep on symmetric geometries:

| Source | Field |
|---|---|
| Long straight wire | $B = \mu_0 I/2\pi r$ |
| Long solenoid | $B = \mu_0 n I$ inside, ~0 outside |
| Toroid | $B = \mu_0 N I/2\pi r$ |

The **Biot-Savart law** handles the rest, integrating contributions $d\mathbf{B} \propto I\,d\mathbf{l}\times\hat{\mathbf{r}}/r^2$ along the wire.

At large distance a current loop looks like a **magnetic dipole** of moment $\mathbf{m} = I\mathbf{A}$, with a field of the same $1/r^3$ shape as the electric dipole's. In a uniform field it feels a torque $\boldsymbol{\tau} = \mathbf{m}\times\mathbf{B}$ but no net force, which is why a compass needle turns rather than being dragged.

> **Common pitfall:** expecting the field to be zero wherever the enclosed current is zero. Ampère's law constrains the *circulation*, not the field. An Amperian loop drawn outside a solenoid encloses equal and opposite currents and gives zero circulation, while the field just inside is large.

## Practice questions

8 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. A long straight wire carries 5.0 A. Compute the magnetic field 0.10 m away, in units of $10^{-6}$ T, using $\mu_0 = 4\pi \times 10^{-7}$ T m/A. Give the answer to one decimal place.

**Answer:** 10 (within ±0.3)

**Why:** $B = \mu_0 I/2\pi r = (4\pi \times 10^{-7} \times 5.0)/(2\pi \times 0.10) = 1.0 \times 10^{-5}$ T $= 10.0 \times 10^{-6}$ T, about a fifth of the Earth’s field.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/a-long-straight-wire-carries-5-0-a-compute-the-magnetic-field-0-10-m

### 2. Sort each consequence by why it holds: because the magnetic field has zero divergence, or because it is the electric field instead.

**Answer:**

- Follows from divergence of B being zero: Magnetic monopoles do not exist, Field lines always close on themselves, Total flux through any closed surface is zero
- True of the electric field instead: Charge acts as a source from which lines begin, A scalar potential can be defined

**Why:** Zero divergence rules out magnetic charge entirely, so lines close and the net flux always vanishes. Sources and a scalar potential belong to electrostatics, and it is precisely the lack of a scalar potential that forces the vector potential on us.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/sort-each-consequence-by-why-it-holds-because-the-magnetic-field-has

### 3. An Amperian loop drawn entirely outside a long solenoid encloses zero net current, so the field just inside the solenoid must be zero.

**Answer:** False

**Why:** False, the loop outside encloses equal and opposite currents from the two sides of the winding, so its *circulation* vanishes. That says nothing about the field elsewhere, and inside the solenoid the field is large and uniform.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/an-amperian-loop-drawn-entirely-outside-a-long-solenoid-encloses-zero

### 4. Which results follow from Ampère’s law applied to symmetric geometries?

A. The field inside a long solenoid is uniform
B. The field of a long straight wire falls as one over the distance
C. The field inside a toroid falls as one over the radius
D. The field of a point charge falls as one over the distance squared

**Answer:** A. The field inside a long solenoid is uniform; B. The field of a long straight wire falls as one over the distance; C. The field inside a toroid falls as one over the radius

**Why:** The wire, solenoid and toroid all follow from Ampère’s law with the right loop. The inverse-square law for a point charge comes from Gauss’s law and has nothing to do with currents.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/which-results-follow-from-amperes-law-applied-to-symmetric

### 5. Complete the comparison between the three symmetric magnetostatic geometries.

**Answer:** The field of a long straight wire falls as one over the **distance**, the field inside a long solenoid is **uniform** across the bore, and the field inside a toroid falls as one over the **radius**, so a toroid, unlike a solenoid, needs a stated **position** before its field means anything.

**Why:** The wire gives $\mu_0I/2\pi r$, the solenoid a uniform $\mu_0 nI$, and the toroid $\mu_0NI/2\pi r$. Because the toroid’s field varies across the bore, quoting it without a radius is meaningless, a distinction that catches people out.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/complete-the-comparison-between-the-three-symmetric-magnetostatic

### 6. A current loop of magnetic moment 0.5 A m$^2$ sits in a 0.4 T field with its moment perpendicular to the field. Set the magnitude of the torque on it, in N m.

**Answer:** 0.2 (within ±0.03)

**Why:** $\tau = mB\sin\theta = 0.5 \times 0.4 \times 1 = 0.2$ N m. The net *force* is zero in a uniform field, opposite sides cancel, so the loop turns without being dragged. A net force needs a field gradient.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/a-current-loop-of-magnetic-moment-0-5-a-m-sits-in-a-0-4-t-field-with

### 7. Match each magnetostatic tool to what it is best used for.

**Answer:**

- Ampère's law → Fields of highly symmetric current arrangements
- Biot-Savart law → Fields of arbitrary current-carrying wires
- Vector potential → A quantity whose curl gives the magnetic field
- Magnetic moment → How a loop behaves far away and in a field

**Why:** Ampère is fast when symmetry allows it, Biot-Savart is general but laborious, the vector potential replaces the impossible scalar one, and the magnetic moment summarises a loop’s distant field and its response to an applied field.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/match-each-magnetostatic-tool-to-what-it-is-best-used-for

### 8. The magnetic field has zero divergence, whereas the static electric field has zero curl. What fundamental mathematical consequence follows for their potentials?

A. The field B is conservative whilst E must be expressed as a curl field
B. Both fields can be written as the gradient of coupled scalar potentials
C. A scalar potential exists for E while B requires a vector potential
D. Neither field admits a potential representation in current-free space

**Answer:** C. A scalar potential exists for E while B requires a vector potential

**Why:** Zero divergence prevents B from being a scalar gradient since curl of gradient vanishes, requiring the vector potential B = curl A. Conversely, irrotational E admits a scalar potential. Non-zero curl means circulation does not vanish, precluding conservative scalar descriptions for B.

Page: https://tryals.app/practice/physics-ii/magnetostatics-in-vacuum/the-magnetic-field-has-zero-divergence-whereas-the-static-electric