Biot–Savart Law: Formula, Applications, and Solved Numericals

The Biot–Savart Law is a fundamental law of electromagnetism in Physics. It helps us in calculating the magnetic field that is produced by a current-carrying conductor.

It plays a similar role to Coulomb's law in electrostatics, but instead of dealing with electric charges, it deals with moving charges that are electric currents.

Biot–Savart Law is one of the important topics because it forms the basis for finding the magnetic field due to current elements, straight conductors, circular loops, and other current configurations.

Requirement of Biot–Savart Law

From Ørsted's experiment, it is noticed that a current-carrying conductor produces a magnetic field around it. However, to calculate the magnitude and direction of this magnetic field at a given point, we require a mathematical relation.

That particular relation is provided by the Biot–Savart Law.

It helps us to find:

How the magnetic field depends on current
How distance affects the magnetic field
How the orientation of the current element affects the field

Statement of Biot–Savart Law

Biot–Savart Law states that:

The magnetic field dB at a point due to a small current element I dl is directly proportional to the current, the length of the current element, and the sine of the angle between the current element and the line joining the element to the point, and inversely proportional to the square of the distance between them.

Mathematical Form of Biot–Savart Law

The magnetic field due to a small current element is given by:

d \vec{B} = \frac{μ_{0}}{4 π} \frac{I d \vec{l} \times \hat{r}}{r^{2}}

The magnitude form is given as:

$d B = \frac{μ_{0}}{4 π} \frac{I d l sin θ}{r^{2}}$

Explanation of Symbols

Symbol	Meaning
dB	Small magnetic field produced
μ₀	Permeability of free space
I	Current in the conductor
dl	Small length element of conductor
θ	Angle between dl and r
r̂	Unit vector along r
r	Distance from current element to point

Value of permeability of free space:

$μ_{0} = 4 π \times 10^{- 7} T·m/A$

Direction of Magnetic Field (Right-Hand Rule)

Determination of the direction of the magnetic field given by the Biot–Savart Law is done by using the right-hand thumb rule.

Point the thumb of the right hand in the direction of the current.
The curl of the fingers gives the direction of the magnetic field lines.

This rule helps us to determine the direction of the dB vector.

Direction of Magnetic Field

Important Observations from Biot–Savart Law

From the formula, we observe:

Magnetic field is always directly proportional to current I.
Magnetic field is always inversely proportional to the square of distance r².
The magnetic field of a current is dependent on the orientation of the conductor.
If θ = 0°, then the magnetic field remains zero.
If θ = 90°, then the magnetic field is maximum.

Applications of Biot–Savart Law

1. Magnetic Field Due to a Long Straight Current-Carrying Conductor

Using the Biot–Savart Law and integration, the magnetic field at a distance r from a long straight wire is given by:

$B = \frac{μ_{0} I}{2 π r}$

2. Magnetic Field at the Centre of a Circular Loop

For a circular loop of radius R carrying current I:

$B = \frac{μ_{0} I}{2 R}$

If the loop consists of N turns, then: $B = \frac{μ_{0} N I}{2 R}$

3. Magnetic Field on the Axis of a Circular Coil

By the use of the Biot–Savart Law, we can calculate the magnetic field at a point on the axis of a circular coil. This application is important in understanding magnetic fields in coils and solenoids.

Limitations of Biot–Savart Law

Applicable only to steady currents
Not suitable for time-varying currents/fields
Calculations become complex for irregular shapes
Cannot be easily applied to closed magnetic circuits

For complex systems, Ampere's Circuital Law is preferred.

Important Points

Use the vector form carefully
Direction matters
θ = 90° gives maximum field
Remember standard results
Units of magnetic field → Tesla (T)

Real-Life Importance

Used in designing electromagnets
Basis of motors and generators
Helps understand magnetic fields in wires
Foundation of modern electromagnetic theory

Summary

The Biot–Savart Law provides a mathematical relationship that is used to calculate the magnetic field produced by a current-carrying conductor. It shows how the magnetic field depends on current, distance, and orientation of the conductor.

Although calculations can be complex, the law is extremely powerful and forms the basis for many standard magnetic field results. Understanding the Biot–Savart Law is essential for mastering magnetism and electromagnetic induction.