Magnetic Field Due to a Current Element, Biot-savart Law

Coulomb's Law perfectly describes the electric field of static charges. But what happens when charges move (i.e., when current flows through a wire)? A new law is needed to calculate the magnetic field produced by such moving charges or current-carrying conductors.

Jean-Baptiste Biot and Félix Savart experimentally established this law in the early 19th century, giving us a precise mathematical tool to calculate B at any point due to any current distribution.

Combining the four dependencies: dB ∝ \[\frac {I dl sin θ}{r^2}\]​

Introducing the constant of proportionality \[\frac {μ_0}{4π}\]​​:

\[dB=\frac{\mu_0}{4\pi}\cdot\frac{Idl\sin\theta}{r^2}\]

\[{d\vec{B}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\hat{r}}{r^2}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\vec{r}}{r^3}}\]

For a finite conductor, integrate over the entire length:

\[{\vec{B}=\frac{\mu_0I}{4\pi}\int\frac{d\vec{l}\times\hat{r}}{r^2}}\]

The magnitude of magnetic induction (dB) at a point due to a small element of current carrying conductor is:
(i) directly proportional to current (dB ∝ I),
(ii) directly proportional to length of element (dB ∝ dl),
(iii) directly proportional to sine of angle between element and line joining its centre to the point (dB ∝ sin θ),
(iv) inversely proportional to square of distance (dB ∝ 1/r²).

Applications

• Magnetic field at centre of circular coil.
• Magnetic field on axis of the coil.
• Magnetic field at a distance from a straight current-carrying conductor.

• The law applies only to steady (constant) currents — not valid for time-varying (alternating) currents
• The current element I\[\overrightarrow {dl}\] must be infinitesimally small; the total field is obtained by superposition (integration)
• Valid in free space (vacuum); for other media, replace μ0​ with μ = μ₀μ_r, where μ_r is the relative permeability
• Obeys the Superposition Principle — total \[\vec B\] due to multiple elements = vector sum of individual contributions

Magnetic Field at the Centre of a Circular Coil

• For a circular coil of radius R, N turns, carrying current I:
  B = \[\frac {μ_0NI}{2R}\] (at centre)
• Direction: Perpendicular to the plane of the coil (use right-hand rule for loop)

Magnetic Field on the Axis of a Circular Coil

• At a point on the axis at a distance x from the centre:
  B = \[\frac{\mu_0IR^2}{2(R^2+x^2)^{3/2}}\]
• At centre (x = 0): reduces to equation (4)
• At large distances (x ≫ R): B ≈ \[\frac{\mu_0IR^2}{2x^3}\]​ (behaves like a magnetic dipole)

Magnetic Field due to a Finite Straight Wire

• At perpendicular distance dd from a finite wire, where ends subtend angles ϕ₁​ and ϕ₂​:
  B = \[\frac {μ_0I}{4πd}\](sin⁡ϕ₁ + sin⁡ϕ₂)
• For infinite straight wire: ϕ₁ = ϕ₂ = 90°, so:
  B = \[\frac {μ_0I}{2πd}\]

Problem Statement

A small current element of length \[d\vec l\] carrying current I is placed along the x-axis at the origin. Find the magnetic field \[d\vec B\] at a point P located on the y-axis at a distance rr from the origin.

Given Information

• Current element: I\[d\vec l\] lies along the +x direction, so \[d\vec l\] = dl \[\ hat x\]
• Point P is on the y-axis, so position vector \[\vec r\] = r\[\hat y\]
• Angle between \[d\vec l\] (+x) and \[\vec r\] (+y): θ=90°

Solution (Step-by-Step)

Step 1: Apply the scalar form of Biot–Savart Law:

Step 2: Substitute θ = 90° → sin ⁡90° = 1:

This is the maximum possible value of dB for a given element, since the point lies on the perpendicular bisector of the element.

Step 3: Find the direction using the cross product:

So \[d\vec B\] points in the +z direction — i.e., out of the xy-plane (out of the page).

Step 4: Write the full vector result: