## 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 (ππ΅β) at a point in space due to a small segment of current (πΌ) is:

**ππ΅β=πβ/4π πΌππβΓπβ/πΒ³**

where:

- ππβ is the infinitesimal length vector of the current element.
- πβ is the position vector from the current element to the point where ππ΅β.
- π is the magnitude of πβ.
- π is the permeability of free space.

## Derivation

### Consider a Current Element:

Assume a small current element ππβ carrying a current πΌ.

### Apply the Concept of Magnetic Field:

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

### 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 ππ΅β is proportional to the current, ππβ, and sinβ‘π (where π*ΞΈ* is the angle between ππβ and πβ), and inversely proportional to πΒ².

These findings form the basis of the Biot-Savart Law: ππ΅β=πβ/4ππΌβππβΓπβ/πΒ³β

## Example

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 π
carrying a current πΌ.

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

### Solution:

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 ππβ on the loop, the position vector to the center of the loop is πβ, and π=π .

Since ππβ is tangential to the loop, πβ is perpendicular to ππβ.

The Biot-Savart Law for this current element becomes: ππ΅β=πβ/4ππΌβππβΓπβ/π3=πβ/4ππΌβππβ/π Β²β

The cross product of ππβ and πβ simplifies because they are perpendicular, and the magnitude becomes ππββ 1.

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

Integrating around the entire loop, the total magnetic field becomes: π΅=πβπΌ/4ππ Β²β 2ππ =πβπΌ/2π β

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

The magnetic field at the center of a circular loop carrying current πΌ with radius π is π΅=πβπΌ/2R. This example shows how the Biot-Savart Law can be applied to find the magnetic field created by specific current distributions.

### Problem**:**

Find the magnetic field at the center of a square current-carrying loop with side length π and current πΌ.

### Solution:

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 βπ/2 to π/2.

The distance from each point on the side to the center is** β**(π/2)Β²+(π/2)Β²=π/**β**2β.

The magnetic field due to a segment ππ₯ is: ππ΅=πβπΌ/4πππ₯/(π/**β**2)Β²β

**Summing Contributions from All Sides:**

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

The result is: π΅=2**β**2πβπΌ/ππβ.

## Practice Problems and Solutions

### Problem 1:

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

### Solution:

**Using Biot-Savart Law:**

The magnetic field at a point on the axis is given by: **ππ΅β=πβ/4ππΌβππβΓπβ/πΒ³**β

ππβ is the small length element, and πβ 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 (π΅π₯β) is given by:** π΅π₯=πβπΌπ
Β²/2(π
Β²+π₯Β²)^3/2**

### Problem 2:

A straight conductor of length πΏ carries a current πΌ. Find the magnetic field at a point π perpendicular to the conductor, at a distance π from its midpoint.

### Solution:

**Setup and Considerations:**

Let the conductor lie along the x-axis from βπΏ/2 to πΏ/2.

Let the point π be along the y-axis at a distance π from the x-axis.

**Applying the Biot-Savart Law:**

The infinitesimal magnetic field due to an element ππ₯ at a distance

π=βπ₯Β²+πΒ²β is: ππ΅=πβπΌππ₯/4ππΒ²

The angle between ππβand πβ is 90β°, making the cross product ππβΓπβ=ππ₯.

**Integrating to Find the Total Field:**

Integrating from βπΏ/2 to πΏ/2, and considering only the perpendicular component: π΅=πβπΌπ/4πβ«βπΏ/2πΏ/2ππ₯(π₯Β²+πΒ²)^3/2β

The integral yields: π΅=πβπΌ/2ππ(πΏ/**β**πΏΒ²+4πΒ²)

### Problem 3:

Find the magnetic field at the center of a square loop of side length π, carrying current πΌ.

### Solution:

**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:

ππ΅=πβπΌ/4πβ«βπ/2π/2ππ₯/(π/2)Β²

**Combining Results:**

After summing the contributions of all four sides: π΅=2**β**2πβπΌ/ππβ