Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law
Introduced| English |
|---|
| permeability/ˌpɜːməˈbɪlɪti/ |
| Biot–Savart law/ˈbɪɒt ˈsævɑːt lɔː/ |
| solenoid/ˈsəʊlənɔɪd/ |
A wire carrying current becomes a magnet
- In 1820 Oersted saw a compass twitch beside a live wire.
- A moving charge — a current — makes its own magnetic field.
- So electricity and magnetism are two sides of one thing.
- This lesson finds the field a current creates.
A current-carrying wire produces its own magnetic field.
Oersted showed a current deflects a compass — it makes a field.
The field circles a straight wire
- Around a long straight wire the field forms circles: $B = \dfrac{\mu_0 I}{2\pi r}$.
- $\mu_0$ is the permeability 磁导率 of free space.
- The field is stronger near the wire and fades as $1/r$.
- Point your right thumb along the current; your fingers curl the way $\vec B$ goes.
The magnetic field around a long straight wire forms:
Field lines circle the wire; $B = \mu_0 I/2\pi r$.
The straight-wire field strength falls off with distance as:
$B = \mu_0 I/2\pi r$ — a $1/r$ falloff.
A wire carries $10\ \text{A}$. At $r = 0.2\ \text{m}$, $B = \mu_0 I/2\pi r$ with $\mu_0 = 4\pi\times10^{-7}$. Find $B$ (in μT).
$B = \dfrac{(4\pi\times10^{-7})(10)}{2\pi(0.2)} = 1\times10^{-5}\ \text{T} = 10\ \mu\text{T}$.
The Biot–Savart law
- To find the field from any shape, add up tiny current bits.
- Each element $I\,d\vec\ell$ makes a small field $d\vec B$ — the Biot–Savart law 毕奥-萨伐尔定律.
- It falls off as $1/r^2$ from each element, like Coulomb's law.
- Integrate over the whole wire to get the total field.
Loops and solenoids concentrate the field
- Bend the wire into a loop and the field lines gather through its centre.
- Stack many loops into a solenoid 螺线管 and the inside field becomes strong and uniform.
- A solenoid behaves just like a bar magnet, with N and S ends.
- More turns or more current makes a stronger electromagnet.
Magnetic field of a wire
The magnetic field around a long straight wire weakens as one over the distance from it.
Many current loops stacked together form a ____, which acts like a bar magnet.
A solenoid concentrates the field into a strong, uniform interior.
Select all true statements about a current's magnetic field.
Circular lines, Biot–Savart integration, solenoid = bar magnet. The field wraps around, not outward.
Fields add as vectors
- When several currents are near, their fields superpose.
- Add the $\vec B$ vectors from each source at the point of interest.
- Two parallel wires can reinforce or cancel between them.
- This superposition is how we build up any real field.
Find the field $0.1\ \text{m}$ from a wire carrying $5\ \text{A}$. ($\mu_0 = 4\pi\times10^{-7}$.)
- $B = \dfrac{\mu_0 I}{2\pi r} = \dfrac{(4\pi\times10^{-7})(5)}{2\pi(0.1)}$.
- $B = 1\times10^{-5}\ \text{T}$, circling the wire.
The straight-wire field falls off as $1/r$ (not $1/r^2$), and it wraps around the wire rather than pointing away from it. Don't picture magnetic field lines shooting outward like electric field lines from a charge — they form closed circles.
A current makes a magnetic field: a straight wire gives circular lines with $B = \mu_0 I / 2\pi r$ (right-hand grip, permeability $\mu_0$). The Biot–Savart law adds up each current element; solenoids concentrate the field into a bar-magnet shape. Fields superpose.