Derivation of Biot Savart Law: Application and Importance

Biot-Savart law states that magnetic field is directly proportional to the length of the conductor and current flowing in the conductor. Derivation of Biot Savart Law depends upon the magnetic properties of the medium and system of the units used. Biot Savart Law relates magnetic field to direction, magnitude, proximity, length of the electric current. This law is used mainly in electromagnetism. 

Key Terms: Biot Savart Law, Coulomb’s law, Electromagnetism, Gauss’s law, Ampere’s Circuital Law, Electric field, Electric current, Conductor, Magnetic field

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Derivation of Biot Savart Law using Point Charges 

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Biot Savart law Derivation for a point charge is mentioned below:

Maxwell’s equation is used for magnetic field and expressing electric field,

\(E = \frac{q}{4\pi\epsilon_0} \times \frac{1 - \frac{v^2}{c^2}}{(1- \frac{v^2 \sin^2\theta}{c^2})^{\frac{3}{2}}} \times \frac{\hat{r'}}{|r'|^2}\)

\(H = v \times D\)

B = \(\frac {1}{c^2}v \times E\)

\(E = \frac{q}{4\pi\epsilon_0} \times \frac{\hat{r'}}{|r'|^2}\)

\(B = \frac{\mu_0q}{4\pi\epsilon_0}v \times \frac{\hat{r'}}{|r'|^2}\)

In the above derivation, q= charged particle and v= constant velocity.

Biot Savart’s Law Video Lecture

Read More: Derivation of Lorentz transformation

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Derivation of Biot Savart Law using Electric Field 

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Derivation of Biot Savart law that is generated by stationary charges used by electric fields, we get

\(\bigtriangledown \times E = 0\)

\(E = - \bigtriangledown \phi\)

Where Φ is the electric scalar potential

The formula that we get from electric fields are created by stationary charges

\(E(r) = \frac{1}{4\pi\epsilon_0} \int \rho(r') \frac{r - r'} {|r - r'|} d^3r' ,\)

Biot Savart law formula that we get before the substitution of j is mentioned below,

\(B(r) =\frac{\mu_0}{4\pi} \int \frac{j(r') * (r - r')}{|r - r'|^3} d^3r',\)

Substituting for j, where dl is element length and I is the vector current

\(j(r)d^3r = I(r) dl,\)

We get Biot Savart expression after substituting for j,

\(B(r) =\frac{\mu_0}{4\pi} \int \frac{I(r') * (r - r')}{|r - r'|^3} dl\)

Also check:

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Derivation of Biot-Savart Law 

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The formula of Biot-Savart Law can be given as follows:

\(dB \propto \frac{Idlsin\theta}{r^2}\) 

or 

\(dB = k\frac{Idl\sin\theta}{r^2}\)

Here, the "K" is the constant that is dependent upon the system of units and magnetic properties of the medium employed. In the unit of the SI system,

\(k = \frac{\mu_0\mu_r}{4\pi}\)

Biot-Savart Law is finally expressed as,

\(dB = \frac{\mu_0\mu_r}{4\pi} \times \frac{Idl\sin\theta}{r^2}\)

Solved Example on Derivation of Biot Savart Law

Example: Let us consider a lengthy wire carrying current 'I' and at a point 'P' in space. Consider a small length infinitely of the 'dl' wire at an r distance from the 'P' point as shown in the below diagram. Here, the distance vector is 'r which makes the θ angle with the current direction in the infinitesimal part of the wire.

Solution: The current through the infinitesimal part of the wire is similar to the current that is carried by the wire itself. It can be expressed by:

\(dB \propto I\)

The magnetic density at the 'P' point is because the infinitesimal length 'dl' of the wire is conversely equivalent to the square of the straight distance from the 'P' point to the 'dl' center. It can be mathematically expressed as: \(dB \propto \frac{1}{r^2}\)

Example of Biot Savart Law

The magnetic field density at the 'P' point due to the infinitesimal part of the wire is equivalent to the real length of the infinitesimal length 'dl' of the wire.

Angle θ is the distance between the direction of current through the infinitesimal part of the wire and the vector r. The 'dl' component of the wire face perpendicular to the 'P' point is dlsinθ. Therefore, 

\(dB \propto dl\sin\theta\)

Now, we can write merging these three statements,

\(dB \propto \frac{Idlsin\theta}{r^2}\)

\(\implies dB = \frac{\mu_0\mu_r}{4\pi}\)

Now, establishing the value of constant K in the above formula, we will get,

Here, μ₀ is the vacuum or the absolute permeability of air used in the expression of constant K. The value is 4π10^-7 Wb/ A^-m in the units of the SI system. μr in the expression of K constant is the relative permeability of the medium.

Now, at the P point, the flux density (B) due to the total length of the wire or current-carrying conductor can be expressed as,

\(B = \int dB = dB = \frac {\mu_0\mu_r}{4\pi} \times \frac{Idl\sin\theta}{r^2} = \frac{\mu_0\mu_r}{4\pi} \int \frac{\sin\theta}{r^2} dl\)

If perpendicular distance D of P point from the wire, then

\(r\sin\theta\) = D or r = \(\frac{D}{\sin\theta}\)

Now, the expression of the B flux density at the P point can be rewritten as,

\(B = \frac{I\mu_0\mu_r}{4\pi} \int \frac{\sin\theta}{r^2} dl = \frac{I\mu_0\mu_r}{4\pi}\int\frac{\sin^3\theta}{D^2}dl\)

\(\frac{l}{D} = \cot\theta \implies I = D\cot\theta\)

Therefore,

\(dl = -D cosec^2 \theta d\theta\)

The formula of B finally comes as,

The angle θ is dependent upon the position of the P point and on the length of the wire. As indicated in the above figure, angle θ for a certain partial length of the wire varies from θ₁ and θ ₂. Due to the total length of the conductor, magnetic flux density at P point is,

Let us consider the wire is long infinitely, then will change from 0 to π i.e., θ₁ = 0 to θ₂ = π

Now, in the final above formula by placing these two values, we will get,

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Applications of Biot Savart law

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The applications of Biot Savart law are mentioned in the below points:

• Biot Savart law is used to evaluate magnetic response at the molecular or atomic level.
• It is used to assess the velocity in aerodynamic theory induced by the vortex line.

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Importance of Biot-Savart Law

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The importance of Biot-Savart Law is mentioned in the below points:

• Biot-Savart Law is exactly similar to Coulomb's law in electrostatics.
• Biot-Savart Law is relevant for very small conductors to carry current.
• For symmetrical current distribution, Biot-Savart Law is applicable.

Also check:

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Previous Year Questions

1. For the magnetic field to be maximum due to a small element of current carrying… [BCECE 2010] 
2. Two long parallel wires carry equal current… [BITSAT 2018]
3. A horizontal overhead powerline is at height of… [BITSAT 2012]
4. ABCD is a square loop made of an uniform conducting wire… [KCET 2001]
5. A current of 2A is flowing in the sides of an equilateral triangle of side… [VITEEE 2015]
6. A current flows in a conductor from east to west… [TS EAMCET 2004]
7. A current I flows in the anticlockwise direction through a square loop of side… [KCET 2016]
8. A (+)vely charged particle is placed near an infinitely long straight conductor… [VITEEE 2016]
9. Two similar coils of radius R are lying concentrically with their planes at… [NEET 2012]
10. The correct Biot-Savart law in vector form is… [KCET 2018]

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Things to Remember

• Biot-Savart law states that magnetic field is directly proportional to the length of the conductor and current flowing in the conductor.
• Biot Savart Law is related to magnetic field and direction, magnitude, proximity, length of the electric current.
• This law is used mainly in electromagnetism. 
• Biot Savart law is relevant to symmetrical current distribution.
• This law is relevant for small conductors to which it carries current.
• It is similar to Coulomb's law in electrostatics.
• Biot Savart law obeys inverse square law like Coulomb's law in electrostatics.