Biot–Savart Law: Definition, Formula, and Applications with Examples

Biot–Savart Law is one of the fundamental laws of electromagnetism, used to calculate the magnetic field produced by a current-carrying conductor. It plays a crucial role in the analysis of magnetic fields in various electrical systems.

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What is Biot–Savart Law?

The Biot–Savart Law relates the magnetic field B produced at a point due to a small segment of current-carrying conductor. This law is analogous to Coulomb’s law in electrostatics but for magnetic fields.

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Biot–Savart Law Formula

The magnetic field d𝐵 at a point due to a small current element Id𝐿 is given by:

$$d\vec{B}=\frac{{\mu }_0}{4\pi }\cdot \frac{Id\vec{L}\times \vec{r}}{r^3}$$

Where:

• μ₀ = Permeability of free space = 4π × 10⁻⁷ T·m/A
• I = Current in the conductor
• dL = Small length element of wire
• r = Position vector from the element to the point of observation
• r (magnitude) = Distance between the current element and the point where the magnetic field is being calculated

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1. Magnetic Field Due to an Infinite Straight Conductor

Configuration:

A straight wire of infinite length carrying current I. Find the magnetic field at a perpendicular distance r from the wire.

Result:

$$B=\frac{{\mu }_{0}I}{2\pi \mathrm{r}}$$

Direction:

Given by the right-hand thumb rule—thumb in direction of current, fingers curl in the direction of magnetic field.

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2. Magnetic Field Due to a Circular Loop

Configuration:

A circular loop of radius R carries current I. Find the magnetic field at the center of the loop.

Result at center:

$$B=\frac{{\mu }_{0}I}{2\mathrm{R}}$$

If the loop has N turns, then:

$$B=\frac{{\mu }_{0}NI}{2\mathrm{R}}$$

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3. Magnetic Field Due to a Circular Arc (Angle θ)

Configuration:

A circular arc of radius R carrying current I subtends angle θ (in radians) at the center.

Result at center:

$$B=\frac{{\mu }_{0}I\theta }{4\pi \mathrm{R}}$$

• For a semicircle (θ = π)
• For a quarter circle (θ = π/2)

4. Magnetic Field Due to a Square Loop

Configuration:

A square loop of side a, carrying current I, find the magnetic field at the center of the square.

Result:

$$B=\frac{2\sqrt{2}{\mu }_{0}I}{\pi a}$$

Each side contributes equally, and symmetry is used to derive the net field at the center.

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Applications of Biot–Savart Law

• Calculation of magnetic fields in motors and generators
• Design of electromagnets and magnetic coils
• Analysis of current-carrying conductors in magnetic circuits
• Electromagnetic field simulation and modeling

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FAQs on Biot–Savart Law

Q1. What is the main limitation of Biot–Savart Law?

It is valid for steady (constant) currents only and not applicable for time-varying fields.

Q2. Is Biot–Savart Law a vector law?

Yes, it gives both magnitude and direction of the magnetic field using vector cross-product.

Q3. How is Biot–Savart Law different from Ampere’s Law?

• Biot–Savart Law is used for calculating field from basic principles (especially in asymmetrical cases).
• Ampere’s Law is used where there is high symmetry (infinite wire, solenoid, toroid).

Q4. What is the direction of the magnetic field using Biot–Savart Law?

It follows the right-hand rule (curling fingers of right hand in direction of integration vector $d\vec{L}$, thumb gives the direction of $\vec{B}$).

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Conclusion

Biot–Savart Law provides a powerful tool to compute the magnetic field generated by any current distribution. From infinite wires to circular and square loops, its applications are critical in understanding electromagnetic theory. Mastery of this law is essential for students and professionals in electrical and electronics engineering, especially those preparing for competitive exams like SSC JE, RRB JE, and GATE.