Biot Savart Law Derivation

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Created by: Team Physics -, Last Updated: May 7, 2024

Biot Savart Law Derivation

The Biot-Savart Law provides a way to calculate the magnetic field generated by a current-carrying conductor. It states that the infinitesimal magnetic field (𝑑𝐡⃗) at a point in space due to a small segment of current (𝐼) is:

𝑑𝐡⃗=πœ‡β‚€/4πœ‹ πΌπ‘‘π‘™βƒ—Γ—π‘Ÿβƒ—/π‘ŸΒ³


  • 𝑑𝑙⃗ is the infinitesimal length vector of the current element.
  • π‘Ÿβƒ— is the position vector from the current element to the point where 𝑑𝐡⃗.
  • π‘Ÿ is the magnitude of π‘Ÿβƒ—.
  • πœ‡ is the permeability of free space.


Consider a Current Element:

Assume a small current element 𝑑𝑙⃗ carrying a current 𝐼.

Apply the Concept of Magnetic Field:

The magnetic field due to this current element at a point 𝑃 at a distance r is perpendicular to both the direction of the current and the line connecting the current element to the point 𝑃.

Calculate the Magnetic Field:

The infinitesimal magnetic field is calculated by considering the contribution of the small current element using experimental observations and the cross product.

Formulate the Biot-Savart Law:

By experimental measurements, it was found that 𝑑𝐡⃗ is proportional to the current, 𝑑𝑙⃗, and sinβ‘πœƒ (where πœƒΞΈ is the angle between 𝑑𝑙⃗ and π‘Ÿβƒ—), and inversely proportional to π‘ŸΒ².

These findings form the basis of the Biot-Savart Law: 𝑑𝐡⃗=πœ‡β‚€/4πœ‹πΌβ€‰π‘‘π‘™βƒ—Γ—π‘Ÿβƒ—/π‘ŸΒ³β€‹


Let’s consider an example of the Biot-Savart law to calculate the magnetic field at the center of a circular current-carrying loop.

Magnetic Field at the Center of a Current-Carrying Loop

Given: A circular loop with radius 𝑅 carrying a current 𝐼.

Find: The magnetic field at the center of the loop.


Place the loop in the xy-plane with its center at the origin.

The current flows in a circular path in the counterclockwise direction.

For an infinitesimal current element 𝑑𝑙⃗ on the loop, the position vector to the center of the loop is π‘Ÿβƒ—, and π‘Ÿ=𝑅.

Since 𝑑𝑙⃗ is tangential to the loop, π‘Ÿβƒ— is perpendicular to 𝑑𝑙⃗.

The Biot-Savart Law for this current element becomes: 𝑑𝐡⃗=πœ‡β‚€/4πœ‹πΌβ€‰π‘‘π‘™βƒ—Γ—π‘Ÿβƒ—/π‘Ÿ3=πœ‡β‚€/4πœ‹πΌβ€‰π‘‘π‘™βƒ—/𝑅²​

The cross product of 𝑑𝑙⃗ and π‘Ÿβƒ— simplifies because they are perpendicular, and the magnitude becomes 𝑑𝑙⃗⋅1.

Since the magnetic field components due to each element 𝑑𝑙⃗ are in the same direction (perpendicular to the loop plane), they add up constructively.

Integrating around the entire loop, the total magnetic field becomes: 𝐡=πœ‡β‚€πΌ/4πœ‹π‘…Β²β‹…2πœ‹π‘…=πœ‡β‚€πΌ/2𝑅​

The factor 2πœ‹ accounts for the total circumference of the loop.

The magnetic field at the center of a circular loop carrying current 𝐼 with radius 𝑅 is 𝐡=πœ‡β‚€πΌ/2R. This example shows how the Biot-Savart Law can be applied to find the magnetic field created by specific current distributions.


Find the magnetic field at the center of a square current-carrying loop with side length π‘Ž and current 𝐼.


The square loop lies in the xy-plane, centered at the origin.

Each side of the square contributes to the magnetic field at the center.

Applying Biot-Savart Law to One Side:

Consider one side of the loop parallel to the x-axis from βˆ’π‘Ž/2 to π‘Ž/2.

The distance from each point on the side to the center is √(π‘Ž/2)Β²+(π‘Ž/2)Β²=π‘Ž/√2​.

The magnetic field due to a segment 𝑑π‘₯ is: 𝑑𝐡=πœ‡β‚€πΌ/4πœ‹π‘‘π‘₯/(π‘Ž/√2)²​

Summing Contributions from All Sides:

The total magnetic field is the vector sum of the contributions from all four sides.

The result is: 𝐡=2√2πœ‡β‚€πΌ/πœ‹π‘Žβ€‹.

Practice Problems and Solutions

Problem 1:

Calculate the magnetic field at a point on the axis of a circular loop of radius 𝑅, carrying a current 𝐼, at a distance π‘₯ from the center of the loop.


Using Biot-Savart Law:

The magnetic field at a point on the axis is given by: 𝑑𝐡⃗=πœ‡β‚€/4πœ‹πΌβ€‰π‘‘π‘™βƒ—Γ—π‘Ÿβƒ—/π‘ŸΒ³β€‹

𝑑𝑙⃗ is the small length element, and π‘Ÿβƒ— is the distance from the element to the point on the axis.

Symmetry Considerations:

The tangential components cancel each other due to symmetry, and only the components along the axis contribute.

The total magnetic field along the axis (𝐡π‘₯​) is given by: 𝐡π‘₯=πœ‡β‚€πΌπ‘…Β²/2(𝑅²+π‘₯Β²)^3/2

Problem 2:

A straight conductor of length 𝐿 carries a current 𝐼. Find the magnetic field at a point 𝑃 perpendicular to the conductor, at a distance π‘Ž from its midpoint.


Setup and Considerations:

Let the conductor lie along the x-axis from βˆ’πΏ/2 to 𝐿/2.

Let the point 𝑃 be along the y-axis at a distance π‘Ž from the x-axis.

Applying the Biot-Savart Law:

The infinitesimal magnetic field due to an element 𝑑π‘₯ at a distance

π‘Ÿ=√π‘₯Β²+π‘ŽΒ²β€‹ is: 𝑑𝐡=πœ‡β‚€πΌπ‘‘π‘₯/4πœ‹π‘ŸΒ²

The angle between 𝑑𝑙⃗and π‘Ÿβƒ— is 90⁰, making the cross product π‘‘π‘™βƒ—Γ—π‘Ÿβƒ—=𝑑π‘₯.

Integrating to Find the Total Field:

Integrating from βˆ’πΏ/2 to 𝐿/2, and considering only the perpendicular component: 𝐡=πœ‡β‚€πΌπ‘Ž/4πœ‹βˆ«βˆ’πΏ/2𝐿/2𝑑π‘₯(π‘₯Β²+π‘ŽΒ²)^3/2​

The integral yields: 𝐡=πœ‡β‚€πΌ/2πœ‹π‘Ž(𝐿/√𝐿²+4π‘ŽΒ²)

Problem 3:

Find the magnetic field at the center of a square loop of side length π‘Ž, carrying current 𝐼.


Analyzing the Problem:

The square loop can be divided into four equal sides.

By symmetry, each side contributes equally to the magnetic field at the center.

Applying the Biot-Savart Law:

Each side contributes a magnetic field perpendicular to the plane of the loop.

For each side, the magnetic field at the center is calculated using the Biot-Savart law:


Combining Results:

After summing the contributions of all four sides: 𝐡=2√2πœ‡β‚€πΌ/πœ‹π‘Žβ€‹

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