Physics · Ch 4 — Moving Charges and Magnetism

Magnetic Field due to a Current Element, Biot-Savart Law

4.4

Magnetic Field due to a Current Element, Biot-Savart Law

Concept: From Current to Magnetic Field

All magnetic fields arise from moving charges (currents) or intrinsic magnetic moments. The Biot-Savart law gives the magnetic field produced by a small, directed piece of current — a current element.

The Law: Vector Form

Consider a conductor carrying a steady current $I$ . Take an infinitesimal element of the conductor, represented by the vector $\mathbf{dl}$ (its direction is the direction of the current). The magnetic field $\mathbf{dB}$ produced by this element at a point $P$ located at a displacement vector $\mathbf{r}$ from the element is:

Proportional to the current $I$ and the length $|\mathbf{dl}|$ .
Inversely proportional to the square of the distance $r = |\mathbf{r}|$ .
Direction: Perpendicular to the plane containing $\mathbf{dl}$ and $\mathbf{r}$ .

In vector form, the law is:

\mathbf{dB} = \frac{\mu_0}{4\pi} \frac{I \, \mathbf{dl} \times \mathbf{r}}{r^3}

Here:

$\mu_0$ is the permeability of free space (or vacuum). Its exact value in SI units is:

\frac{\mu_0}{4\pi} = 10^{-7} \, \text{T m/A} \quad \text{or} \quad \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}

The cross product $\mathbf{dl} \times \mathbf{r}$ gives the direction (right-hand screw rule: curl fingers from $\mathbf{dl}$ to $\mathbf{r}$ ; thumb gives direction of $\mathbf{dB}$ ).
This expression holds for a current element in vacuum.

The Law: Magnitude Form

Using the property of the cross product, $|\mathbf{dl} \times \mathbf{r}| = dl \, r \sin\theta$ , where $\theta$ is the angle between $\mathbf{dl}$ and $\mathbf{r}$ . The magnitude of the magnetic field is:

|\mathbf{dB}| = \frac{\mu_0}{4\pi} \frac{I \, dl \, \sin\theta}{r^2}

Key point: When $\theta = 0$ (point $P$ lies along the line of the current element), $\sin\theta = 0$ , so $\mathbf{dB} = 0$ . The field is zero directly in front of or behind the element.

Comparison with Coulomb’s Law

Feature	Biot-Savart Law (Magnetic)	Coulomb’s Law (Electric)
Source	Vector source: $I\,\mathbf{dl}$	Scalar source: charge $q$
Distance dependence	$\propto 1/r^2$	$\propto 1/r^2$
Direction of field	Perpendicular to $\mathbf{r}$ and $\mathbf{dl}$	Along $\mathbf{r}$ (radial)
Angle dependence	Yes ( $\sin\theta$ )	No
Superposition	Applies (linear in source)	Applies (linear in source)

Connection Between Constants

The constants $\mu_0$ (permeability) and $\epsilon_0$ (permittivity) are related to the speed of light $c$ :

\mu_0 \epsilon_0 = \frac{1}{c^2}

Since $c = 3 \times 10^8$ m/s is fixed, choosing $\mu_0$ fixes $\epsilon_0$ (and vice versa). In SI, $\mu_0$ is defined as $4\pi \times 10^{-7}$ .

Worked Example (from textbook) …

Figure 4.7Illustration of the Biot-Savart law. The current element I dl produces a field dB at a distance r. The ⊗ sign indicates that the field is perpendicular to the plane of this page and directed into it.

Fig. 4.7 — Illustration of the Biot-Savart law. The current element I dl produces a field dB at a distance r. The ⊗ sign indicates that the field is perpendicular to the plane of this page and directed into it.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your NCERT textbook's own diagram.

What the Figure Shows

The diagram depicts a finite curved conductor XY carrying a steady current $I$ flowing upward (from X at the bottom to Y at the top). At a chosen point on the conductor, a bold arrow labeled "Current element" represents the infinitesimal vector $I \, d\mathbf{l}$ — its direction is tangent to the wire, pointing along the current. A dashed line extends along this element to show its line of action.

From the same point, a blue arrow labeled $\mathbf{r}$ (the displacement vector) points from the element to a point P located to the right. The angle $\theta$ between the direction of $d\mathbf{l}$ and $\mathbf{r}$ is marked at the element. At point P, a ⊗ symbol (cross inside a circle) labeled $d\mathbf{B}$ indicates that the magnetic field due to the element is directed into the page (perpendicular to the plane of the diagram).

Physical Idea Taught

The figure illustrates the Biot-Savart law, which gives the magnetic field $d\mathbf{B}$ produced by a tiny segment $I\,d\mathbf{l}$ of a current-carrying conductor. The key idea is that the field at a point depends on:

The current $I$ and the length $|d\mathbf{l}|$ of the element.
The inverse square of the distance $r$ from the element to the point.
The sine of the angle $\theta$ between $d\mathbf{l}$ and $\mathbf{r}$ .
The direction of $d\mathbf{B}$ is perpendicular to the plane containing $d\mathbf{l}$ and $\mathbf{r}$ , given by the right-hand screw rule.

The ⊗ symbol at P shows that for the geometry shown (with $d\mathbf{l}$ upward and $\mathbf{r}$ to the right), the field points into the page.

Key Formula Developed

The Biot-Savart law in vector form is:

d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{r^3}

where:

$\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$ is the permeability of free space.
$I$ is the current in the conductor.
$d\mathbf{l}$ is the infinitesimal length vector of the current element (direction of current).
$\mathbf{r}$ is the displacement vector from the element to the field point. …

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