title: “3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)”

aliases:

• Biot-Savart Law
• Finite Straight Wire Derivation
• Circular Loop Field Derivation
• Biot-Savart vs Ampere Law

tags:

• ece2105
• electromagnetics
• magnetostatics
• biot-savart
• study-order/3.02

type: uni-note

course: “ECE 2105 - Electromagnetic Fields and Waves”

chapter_map: “03 Chapter Map - Magnetostatics & Magnetic Materials”

term: 2-1

teacher: Mashuk Sir / Sanglap Sir

status: enhanced

---

Related Concepts: 3.01 Fundamental Postulates of Magnetostatics & Lorentz Force Equation | 3.03 Ampere’s Circuital Law & Vector Magnetic Potential (A) | 3.04 Magnetization, Magnetic Materials, Boundary Conditions & Hall Effect

3.02 Biot-Savart Law Applications (Finite Wire, Circular Loop, Coaxial Cable)

Overview

The Biot-Savart Law calculates the magnetic flux density dB produced at a point in space by a differential current element I dl’.

1. Statement & Vector Formula of Biot-Savart Law [PYQ: 2015, 2018, 2023, 2025]

dB = (u0 I / 4pi) * (dl’ x aR / R^2) ⇒ B = (u0 I / 4pi) * Integral (dl’ x aR / R^2)

1.1 Biot-Savart Law vs. Ampere’s Circuital Law Comparison [PYQ: 2015, 2018, 2023, 2025]

Feature	Biot-Savart Law	Ampere’s Circuital Law
Mathematical Method	Direct vector integration over current distribution.	Line integral along closed path C (Integral H . dl = I).
Geometry Requirement	General. Works for asymmetric wires and arbitrary loops.	Practical only for highly symmetric systems.
Electrostatic Analogy	Analogous to Coulomb’s Law.	Analogous to Gauss’s Law.

2. Master Derivation 1: Finite Straight Wire of Length 2L [PYQ: Heavily Tested: 2015, 2016, 2021, 2023, 2025]

Let a straight wire of length 2L lie along the z-axis from z = -L to z = +L, carrying current I. Find B at observation point P(r, 0, 0) at distance r on the x-axis.

 Z-axis

+L (Wire Top)

|

| R = sqrt(r^2 + z’^2)

| /

dz’ ----+(z’) -------⇒ P(r, 0, 0) [Observation Point at distance r]

| \

| \

-L (Wire Bottom)

2.1 Step-by-Step Integration

1. Current Element: dl’ = az dz’. Distance vector R = ar r - az z’.
2. Cross Product: dl’ x R = (az dz’) x (ar r - az z’) = aphi r dz’.
3. Integral Setup: B = aphi (u0 I r / 4pi) * Integral from -L to +L of dz’ / (r^2 + z’^2)^(3/2)
4. Evaluate Integral (Trigonometric Substitution z’ = r tan theta): Integral = 2L / (r^2 * sqrt(r^2 + L^2))
5. Final Result:

B = aphi (u0 I L) / (2pi r sqrt(r^2 + L^2))

Infinite Wire Limiting Case (L → infinity) [PYQ: 2015, 2021, 2025]

When L >> r, L / sqrt(r^2 + L^2) → 1: B = aphi (u0 I / 2pi r)

3. Master Derivation 2: Circular Current Loop along Axis [PYQ: 2016, 2018, 2020, 2022]

For a circular current loop of radius b in the xy-plane carrying current I, the magnetic flux density at height z along the z-axis is:

B = az (u0 I b^2) / (2 (b^2 + z^2)^(3/2))

• At Center (z=0): B = az (u0 I / 2b)