---
title: "Biot-Savart vs Ampere's law, coil axis"
subject: "Engineering Physics"
chapter: 8
key: "semester_1/engineering_physics:biot-ampere-compare-coil-axis"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/biot-ampere-compare-coil-axis
site: BCT Engineering
---

# Biot-Savart vs Ampere's law, coil axis

Engineering Physics · Chapter 8: Electromagnetism

- Asked in: 2070 Asar, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/biot-ampere-compare-coil-axis
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-8

## Question

Compare the methods of Biot and Savart Law and Ampere's Law to calculate magnetic fields due to current carrying conductor. Calculate magnetic field at an axial distance 'x' from the center of the circular coil carrying current.

## Short answer

Biot-Savart law handles any current geometry by integrating dB = (μ₀/4π) I dl × r̂ / r²; Ampere’s law ∮B·dl = μ₀ I_enclosed requires symmetry. On the axis of a circular coil of N turns, radius R, current I, at distance x, B = μ₀ N I R² / [2 (R² + x²)^(3/2)].

## Answer

#### Comparison

| Point | Biot-Savart law | Ampere's law |
|---|---|---|
| Form | dB = (μ₀/4π) I dl × r̂ / r² | ∮B·dl = μ₀ I_enc |
| Idea | Field of each small element, then add up | Line integral of B round a closed loop |
| Symmetry needed | Not needed | Needed (B constant along the loop) |
| Use | Any shape of conductor | Infinite wire, solenoid, toroid, sheet |
| Maths | Integration, often hard | Often just algebra |
| Analogy | Coulomb's law | Gauss's law |

Ampere's law is quicker when symmetry exists. Biot-Savart works for every current distribution.

#### Field on the axis of a circular coil
Coil of radius R, N turns, current I. Point P is at distance x on the axis.

```text
     I dl (element on the ring)
      o
      | \
    R |   \  r = √(R² + x²)
      |     \
      O-------P-----> x (axis)
          x
   (ring seen edge-on, centre O)
```

1. Take a current element I dl on the ring. Its distance from P is r = √(R² + x²), and dl ⟂ r, so dB = (μ₀/4π) I dl / r².
2. By symmetry the components of dB perpendicular to the axis cancel in pairs. Only the axial component of dB survives. It equals dB sin α, where sin α = R/r.
3. dB_axial = dB sin α = (μ₀ I dl / 4π r²)(R/r).
4. Integrate over the ring, ∮dl = 2πR:
   B = (μ₀ I R/4π r³)(2πR) = μ₀ I R²/(2 r³).
5. For N turns:

**B = μ₀ N I R² / [2 (R² + x²)^(3/2)]**

At the centre (x = 0), B = μ₀NI/(2R). Far away (x ≫ R), B ≈ μ₀NIR²/(2x³).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which statement correctly compares the two laws?
   A. Ampere's law needs symmetry (B constant on the loop); Biot-Savart works for any conductor shape
   B. Biot-Savart needs symmetry; Ampere's law works for any shape
   C. Both need high symmetry
   D. Neither needs symmetry
   Answer: A. Ampere's law is quick only when symmetry exists; Biot-Savart applies to every current distribution. (Key point: Symmetry need)

2. On the axis of a circular coil, which part of dB from each element survives integration?
   A. The component perpendicular to the axis
   B. The component along the axis, dB sin α
   C. The full dB with no cancellation
   D. Neither; the field is zero on the axis
   Answer: B. Perpendicular components cancel in pairs by symmetry. (Key point: Axial component)

3. The field on the axis of an N-turn coil of radius R at distance x is:
   A. B = μ₀NIR/[2(R² + x²)]
   B. B = μ₀NI R²/[2 (R² + x²)^(1/2)]
   C. B = μ₀NIR²/[2(R² + x²)^(3/2)]
   D. B = μ₀NI/(2x)
   Answer: C. sin α = R/r with r = √(R²+x²), and ∮dl = 2πR give this result. (Key point: Axis formula)

4. At the centre of the coil (x = 0) the field is:
   A. μ₀NI R²/2
   B. zero
   C. μ₀NI/(4πR)
   D. μ₀NI/(2R)
   Answer: D. Putting x = 0 gives μ₀NIR²/(2R³) = μ₀NI/(2R). (Key point: Limits)

## Related in Engineering Physics

- [Magnetic field on axis of circular coil](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acircular-coil-axis-field) — Engineering Physics, chapter 8, asked 2×
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- [Induced magnetic field, displacement current](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisplacement-current-induced-field-circular-plates) — Engineering Physics, chapter 9

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