## What is Biot-Savart Law?

Biot-Savart’s law is an equation that gives the magnetic field produced due to a current carrying segment. This segment is taken as a vector quantity known as the current element.

## Biot-Savart’s Law Formula

Consider a current carrying wire ‘i’ in a specific direction as shown in the above figure. Take a small element of the wire of length ds. The direction of this element is along that of the current so that it forms a vector i ds.

To know the magnetic field produced at a point due to this small element, one can apply Biot-Savart’s Law. Let the position vector of the point in question drawn from the current element be **r** and the angle between the two be θ. Then,

\(\left | dB \right |=(\frac{\mu _{0}}{4\pi })(\frac{Idlsin\Theta }{r^{2}})\) |

Where

- μ
_{0}is the permeability of free space and is equal to 4π × 10^{-7}TmA^{-1}.

The direction of the magnetic field is always in a plane perpendicular to the line of element and position vector. It is given by the right-hand thumb rule where the thumb points to the direction of conventional current and the other fingers show the magnetic field’s direction.

In the figure shown above, the direction of the magnetic field is pointing into the page.

This can be expressed in terms of vectors as:

d→B = μ_{0}4π i →ds ×^rr2

Let us use this law in an example to calculate the Magnetic field due to a wire carrying current in a loop.

**Example of Biot-Savart’s Law**

**The magnetic field of Current Loop:**

Consider a current loop of radius R with a current ‘i’ flowing in it. If we wish to find the electric field at a distance l from the center of the loop due to a small element ds, we can use the Biot-Savart Law as:

d→B = μ_{0}4π i d→s ×^rr2

Consider the current element ids at M which is coming out of a plane in the figure. Since r is in the plane of the page, the two of them are perpendicular to each other. Furthermore, the magnetic field produced db is also in the plane of the page.

dB = μ_{0}4π i ds . 1. sin 90⁰r2 = μ_{0}4π i dsr2

But from the figure,

R2 + l2 = r2

dB = μ_{0}4π i dsR2 + l2

Now, if we consider the diametrically opposite element at N, it produces a field such that it’s component perpendicular to the axis of the loop is opposite to that of the field produced at M. Thus only the axial components remain. We can divide the loop into diametrically opposite pairs and apply the same logic.

Also note that from figure that

α = θ

∴ cos θ = R√R2 + l2

Thus,

dB cos θ = μ_{0}4π i dsR2 + l2 × R√R2 + l2

The total field will be thus,

B = ∫ μ_{0}4π i ds R(R2 + l2)32 = μ_{0}4π i R(R2 + l2)32 ∫ ds

B = μ_{0}4π i R(R2 + l2)32 × 2πR

B = μ_{0} i R22(R2 + l2)32

The right hand thumb rule can be used to find the direction of magnetic field.

### Applications of Biot-Savart’s Law

Some of Biot-Savart’s Law applications are given below.

- We can use Biot–Savart law to calculate magnetic responses even at the atomic or molecular level.
- It is also used in aerodynamic theory to calculate the velocity induced by vortex lines.

If there are ‘n’ loops carrying current in the same direction placed very close to one another, can you calculate the field using Biot-Savart’s Law? What if we placed a loop carrying current in the opposite direction right next to the loop in the example we demonstrated? Will it still produce a magnetic field or behave like an electric dipole only this time the dipole is magnetic? Stay tuned with BYJU’S to learn more on these.