Biot Savart Law: Statement, Derivation An Applications

What is Biot Savart Law

The Biot Savart Law gives the magnetic field produced by a steady electric current. Each current element contributes a vector field set by its magnitude, direction and displacement from the observation point. The law is consistent with Ampere’s circuital law and Gauss’s theorem for magnetism in magnetostatics. It plays a role in magnetostatics similar to Coulomb’s law for electrostatic charge distributions.

French physicists Jean-Baptiste Biot and Félix Savart reported the relationship in 1820. The Biot-Savart law expresses the differential magnetic flux density due to a current element. Its direction is perpendicular to both the element direction and the displacement towards the field point.

The magnitude dB is proportional to current I, current-element length dl and sin θ, where θ is the angle between the current element and the displacement vector. Its distance dependence follows 1/r². The cross product in the vector form supplies the field direction.

Biot Savart Law Statement & Derivation

The differential form of the Biot-Savart law is:

The constant k depends on the unit system and permeability used in the model. In the SI system of unit, the free-space factor is:

The corresponding Biot-Savart law derivation for a line-current element is:

Consider a straight wire carrying steady current I and an observation point P. Select an infinitesimal directed wire element dl. The displacement vector r points from that element to P and makes angle θ with dl.

The contribution dB at P is proportional to the current I carried by the element.

Because the same steady current passes through each series element of the wire, this proportionality is:

The contribution from one element also decreases with the square of its distance from P:

The magnetic field contribution is proportional to the element length dl.

Only the component of dl perpendicular to r contributes to the cross product. Its magnitude is dl sin θ.

Combining the current, length, angle and distance factors gives:

This is the scalar magnitude form of Biot Savart’s Law.

Substituting the SI free-space factor for k gives:

Here, μ₀ is the vacuum absolute permeability. Since the 2019 SI revision, μ0 has been experimentally determined rather than fixed exactly; 4π × 10^-7 H/m remains an excellent approximation. H/m is equivalent to W_b/(A·m). For a homogeneous linear isotropic medium, μ = μ0μr may be used, where μr is the relative permeability. More general magnetic media require their constitutive relation rather than a single constant μr.

The total flux density B at P is the vector integral of contributions from the complete current-carrying conductor:

Let D be the perpendicular distance from point P to the straight wire. The geometry gives:

Using that relation, the integral for B becomes:

The wire geometry also gives the following substitution:

After integration, B is:

The endpoint angles depend on wire length and the position of P. For a finite segment, let θ vary from θ1 to θ2. The magnetic flux density due to the entire conductor is:

For an infinitely long wire, the endpoints approach θ₁ = 0 and θ₂ = π in the angle convention used above. Substitution in the finite-wire result from the Biot Savart law gives:

The same infinite-wire result follows directly from Ampere’s Law because cylindrical symmetry makes B constant in magnitude around a circular Amperian path.