Step-by-Step Derivation of Biot Savart’s Law

The Biot-Savart law gives the mathematical expression for the magnetic field produced by a small current element at a given point in space. Its derivation is fundamental in understanding the magnetic effects of electric currents, particularly in problems related to current-carrying conductors and their magnetic fields.

Fundamental Statement of Biot-Savart Law

The Biot-Savart law states that the magnetic field $d\vec{B}$ at a point due to a current element $I d\vec{l}$ is directly proportional to the current, the length of the element, and the sine of the angle between the element and the line joining it to the observation point, and inversely proportional to the square of the distance.

The law is mathematically represented as:

$d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I \, d\vec{l} \times \vec{r}}{r^3}$

Here, $d\vec{B}$ is the magnetic field, $I$ is the current, $d\vec{l}$ is the length vector of the current element, $\vec{r}$ is the position vector from the current element to the observation point, and $r$ is the magnitude of $\vec{r}$.

For detailed explanation of the Biot Savart law, refer to Biot Savart Law Explained.

Stepwise Derivation of Biot-Savart Law

The derivation of Biot-Savart law from basic principles relies on experimental observations and the use of proportionality constants. Consider a straight, small current element $I d\vec{l}$ placed in a medium.

The magnetic field produced at point $P$ at a distance $r$ is found to satisfy the following properties, verified by experiment:

• Proportional to current $I$ in the conductor
• Proportional to the element length $|d\vec{l}|$
• Proportional to $\sin\theta$, where $\theta$ is the angle between $d\vec{l}$ and $\vec{r}$
• Inversely proportional to $r^2$

These observations yield the relation:

$dB \propto \dfrac{I\, |d\vec{l}|\, \sin\theta}{r^2}$

To convert this into an equation, introduce the proportionality constant $k$:

$dB = k \cdot \dfrac{I\, |d\vec{l}|\, \sin\theta}{r^2}$

In SI units, $k = \dfrac{\mu_0}{4\pi}$, where $\mu_0$ is the permeability of free space with value $4\pi \times 10^{-7} \, T \, m/A$.

From vector analysis, $|d\vec{l}| \sin\theta$ equals the magnitude of the cross product $|d\vec{l} \times \hat{r}|$, so Biot-Savart law in vector form is:

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

Here, $\hat{r} = \dfrac{\vec{r}}{r}$ is the unit vector from the current element to the point of observation.

For complete understanding of the magnetic effects produced by currents, refer to Magnetic Effects of Current.

Derivation of Biot-Savart Law from Ampere's Law

Ampere’s law relates the magnetic field in space to the current producing it through $ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}} $. For a straight, thin, infinitely long wire, the Biot-Savart law can be derived as a special case from Ampere’s circuital law.

Consider a straight current-carrying conductor. Applying Ampere’s law for a circular path of radius $r$ around the wire gives $B (2\pi r) = \mu_0 I$. Thus, $B = \dfrac{\mu_0 I}{2\pi r}$. The infinitesimal magnetic field $d\vec{B}$ due to a small element $d\vec{l}$ can be found using superposition from all such elements, arriving at the Biot-Savart expression.

Biot-Savart Law: Vector Form

In vector notation, the magnetic field $d\vec{B}$ at point $P$ due to a current element is expressed as:

$d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I \, d\vec{l} \times \vec{r}}{r^3}$

This compact vector form shows both the direction and magnitude of the field. The cross product indicates the direction perpendicular to both $d\vec{l}$ and $\vec{r}$, as determined by the right-hand rule.

Applications of Biot-Savart Law

The Biot-Savart law is used to compute the magnetic field for various current configurations such as straight wires, circular loops, and solenoids. It is essential for evaluating the magnetic field at any spatial point given a steady current distribution.

• Determining field due to a straight current-carrying wire
• Calculating magnetic field at the center of a circular current loop
• Evaluating fields from complex current arrangements

To practice application-based questions, access Electromagnetic Induction Mock Test.

Tabular Summary: Key Aspects of Biot-Savart Law

Key Features and Proof of Biot-Savart Law

The Biot-Savart law provides a stepwise approach to constructing the total magnetic field due to any current distribution by integrating the effects of infinitesimal current elements.

The law can be proven experimentally by examining the field pattern and its dependence on current and distance, further confirmed by deduction from Maxwell’s equations in advanced studies.

Further Exploration and Practice

A comprehensive understanding of the Biot-Savart law extends to its role in electromagnetic waves and advanced current distributions.

For additional practice, consult Electromagnetic Induction Practice Paper.

To relate the Biot-Savart law with wave phenomena, refer to Introduction to Electromagnetic Waves.