Magnetic Field Due to a Current Element – Biot-Savart Law 🧲

Every magnetic field you ever meet comes from moving charge – either a flowing current or the tiny “built-in” currents inside particles. The Biot-Savart law tells you exactly how a short piece of current-carrying wire creates that field.

1. Vector Form (the whole story)

\[ d\mathbf B \;=\;\frac{\mu_0}{4\pi}\; \frac{I\,d\boldsymbol\ell \times \mathbf r}{r^{3}} \] :contentReference[oaicite:0]{index=0}

  • I is the current through the element.
  • dℓ is a tiny length pointing with the current.
  • r is the arrow from the element to the point where you want d𝐁.
  • d𝐁 points ⟂ to the plane made by dℓ and 𝐫 (use the right-hand screw 🤚🔄).

2. Magnitude Form (easy to plug numbers)

\[ dB \;=\;\frac{\mu_0}{4\pi}\; \frac{I\,|d\ell|\,\sin\theta}{r^{2}} \] :contentReference[oaicite:1]{index=1}

Here θ is the angle between the wire bit and 𝐫. When θ = 0°, d𝐁 vanishes – field is zero straight ahead of the element.

3. The Magic Constant

\[ \frac{\mu_0}{4\pi}=10^{-7}\,\text{T·m A}^{-1} \] (μ0 is the permeability of free space) :contentReference[oaicite:2]{index=2}

4. Spot the Differences (and Twins!) with Electric Fields ⚡

  • Both magnetic and electric fields fade like 1 / r2 and obey superposition. :contentReference[oaicite:3]{index=3}
  • Electric field springs from a scalar charge; magnetic field comes from the vector current element I dℓ. :contentReference[oaicite:4]{index=4}
  • Electric field points along 𝐫; magnetic field sits ⟂ to 𝐫 and dℓ. :contentReference[oaicite:5]{index=5}
  • Magnetic field carries an extra sin θ factor, so angle matters! :contentReference[oaicite:6]{index=6}

5. A Speed-of-Light Connection 🌟

\[ \mu_0\,\varepsilon_0=\frac{1}{c^{2}} \] :contentReference[oaicite:7]{index=7}

This neat link shows how electric and magnetic “constants” lock together to set the ultimate speed limit c.

6. Worked Example 🚀

An element of wire 1 cm long lies along the x-axis at the origin, carrying 10 A. What field do you get 0.5 m up the y-axis?

  1. Angle θ = 90°, so sin θ = 1.
  2. \[ dB=\frac{\mu_0}{4\pi}\; \frac{I\,\Delta x\,\sin\theta}{r^{2}} =\frac{10^{-7}\times10\times0.01}{(0.5)^{2}} =4\times10^{-8}\,\text{T} \] :contentReference[oaicite:8]{index=8}
  3. Right-hand rule gives +z direction (out of the page 👍).

7. Quick Right-Hand Reminder 🤚

Sweep your right fingers from dℓ toward 𝐫; your thumb pops along d𝐁. If your sweep is anticlockwise, field points toward you; clockwise, it dives away. :contentReference[oaicite:9]{index=9}

High-Yield NEET Nuggets 🔑

  • The complete vector law \(d\mathbf B = (\mu_0/4\pi)\,I\,d\boldsymbol\ell \times \mathbf r\,/ r^{3}\).
  • Angle factor sin θ — field is zero along the wire’s own direction.
  • Numerical value \( \mu_0/4\pi = 10^{-7}\,\text{T·m A}^{-1}\) — an easy plug-in for calculations.
  • Right-hand screw rule for direction of the tiny field.
  • Contrast with Coulomb’s law: scalar vs vector source, parallel vs perpendicular field directions.