# Magnetic Field and Ampere's Law

Physics I · Principles of Electromagnetism and Optics · https://tryals.app/learn/physics-i/magnetic-field-and-amperes-law

## Magnetic Field Basics

Magnetism is electricity's velocity-dependent twin: it acts only on **moving** charges, always at right angles to their motion.

The **magnetic force** on a charge is given by $\vec{F} = q\vec{v}\times\vec{B}$, where $q$ is charge, $\vec{v}$ is velocity, and $\vec{B}$ is the magnetic field.

Because this force is always perpendicular to $\vec{v}$, it changes direction but never speed or kinetic energy. If a problem needs energy added, an electric field must do it.

The **Biot-Savart law** gives the field from a current element:

$d\vec{B} = \frac{\mu_0}{4\pi}\frac{I\,d\vec{l}\times\hat{r}}{r^2}$

Here $\mu_0$ is the permeability of free space, $I$ is current, $d\vec{l}$ is length, and $r$ is distance.

## Ampere's Law and Results

**Ampere's law** relates magnetic fields to steady currents with symmetry:

$\oint_C \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$

| Source | Field Formula | Notes |
| --- | --- | --- |
| Long straight wire | $B = \dfrac{\mu_0 I}{2\pi r}$ | Distance $r$ |
| Solenoid (inside) | $B = \mu_0 n I$ | Turns per length $n$ |
| Toroid (inside) | $B = \dfrac{\mu_0 N I}{2\pi r}$ | Total turns $N$ |

The **force between parallel wires** carrying currents $I_1$ and $I_2$ is $F/L = \dfrac{\mu_0 I_1 I_2}{2\pi d}$. Parallel currents attract; antiparallel repel.

**Physics link:** A charge in a uniform $\vec{B}$ moves in a circle of radius $r = mv/(qB)$, used in cyclotrons.

## Practice questions

5 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. An electron moves through a uniform magnetic field. The magnetic force is always **perpendicular** to its velocity. What can this force change?

A. Neither: perpendicular forces do nothing
B. Both speed and direction
C. The direction of motion only
D. The speed only, never the direction

**Answer:** C. The direction of motion only

**Why:** A perpendicular force does no work, so the kinetic energy, and hence speed, is frozen. All the force can do is bend the path, which is why charges in uniform magnetic fields travel in circles and spirals rather than speeding up.

Page: https://tryals.app/practice/physics-i/magnetic-field-and-amperes-law/an-electron-moves-through-a-uniform-magnetic-field-the-magnetic

### 2. Magnetic field lines always form closed loops (they have no beginning or end).

**Answer:** True

**Why:** $\nabla \cdot B$ = 0: no magnetic monopoles exist, so field lines are always closed.

Page: https://tryals.app/practice/physics-i/magnetic-field-and-amperes-law/magnetic-field-lines-always-form-closed-loops-they-have-no-beginning

### 3. Complete the statement with the correct term.

**Answer:** The force on a moving charge in a magnetic field is $F = qv \times B$. This is called the **Lorentz** force.

**Why:** The Lorentz force $F = q(E + v \times B)$ includes both electric and magnetic contributions.

Page: https://tryals.app/practice/physics-i/magnetic-field-and-amperes-law/complete-the-statement-with-the-correct-term

### 4. Magnetic field lines always close on themselves while electric field lines begin and end on charges. What does that difference record?

A. That magnetic forces do no work on moving charges
B. That no magnetic monopole has been found
C. That magnetic fields are produced only by currents and not by charges
D. That magnetic fields are always stronger than electric fields

**Answer:** B. That no magnetic monopole has been found

**Why:** Closed loops are the geometric expression of no monopoles. Cut a magnet and you get two magnets, never an isolated pole, the field lines have been telling you that all along.

Page: https://tryals.app/practice/physics-i/magnetic-field-and-amperes-law/magnetic-field-lines-always-close-on-themselves-while-electric-field

### 5. Which affect the magnetic force on a charged particle?

A. Charge magnitude
B. Angle between v and B
C. Speed of the particle
D. Mass of the particle

**Answer:** A. Charge magnitude; B. Angle between v and B; C. Speed of the particle

**Why:** $F = qvB \sin \theta$ depends on charge, speed, and angle, not mass.

Page: https://tryals.app/practice/physics-i/magnetic-field-and-amperes-law/which-affect-the-magnetic-force-on-a-charged-particle