---
title: "Electron on ring axis performs SHM"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:ring-electron-shm-axial"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/ring-electron-shm-axial
site: BCT Engineering
---

# Electron on ring axis performs SHM

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2072 Kartik, 5 marks
- Video: [If an electron is placed along the axis of a uniformly charged circular ring ... #neet #jee #cbse](https://www.youtube.com/watch?v=IGhc-BRBuIU) — Physics : easy peasy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/ring-electron-shm-axial
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

A thin ring made of plastic of radius R is uniformly charged with linear charge density λ. Calculate the electric field intensity at any point at an axial distance y from the center. If an electron is constrained to be in the axial line of the same ring, show that the motion of the electron is SHM.

## Short answer

The axial field is E = λ R y / [2ε₀ (R² + y²)³ᐟ²]. For small y ≪ R, E ≈ λ y/(2ε₀ R²); the restoring force F = −eE ∝ −y gives SHM with period T = 2π √(2ε₀ m R²/(eλ)).

## Answer

#### Field on the axis of a charged ring
Ring of radius R, linear charge density λ (total Q = 2πRλ). Point P on the axis at distance y from the centre.

```text
     dq (element on the ring)
      o
      | \
    R |   \  s = √(R² + y²)
      |     \
      O-------P-----> axis
          y
   (ring seen edge-on, centre O)
```

1. Take element dq = λ dl. Its distance from P is s = √(R² + y²). dE = (1/4πε0) dq / s².
2. Components perpendicular to the axis cancel by symmetry. The axial component is dE cosφ, with cosφ = y/s.
3. E = ∫ (1/4πε0) (dq/s²)(y/s) = (1/4πε0) · y Q / (R² + y²)^(3/2)

**E = λ R y / (2ε0 (R² + y²)^(3/2))**, directed along the axis away from the ring (λ > 0).

#### SHM of an electron
1. For λ > 0, the force on the electron (charge −e) is towards the centre: F = −eE.
2. For small displacements y ≪ R, (R² + y²)^(3/2) ≈ R³, so

E ≈ λ y / (2ε0 R²), and F = −(eλ / 2ε0R²) y = −K y

3. F is proportional to the displacement and opposite to it: this is the condition for simple harmonic motion.
4. m d²y/dt² = −K y → ω = √(K/m) = √(eλ / (2ε0 m R²))

**T = 2π √(2ε0 m R² / (eλ))**

So the electron oscillates simple-harmonically about the centre of the ring, for small amplitudes.

## Check yourself

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

1. On the axis of a ring, which part of the element's field adds up?
   A. dE sin φ with sin φ = R/s
   B. The full dE with no angle factor
   C. dE cos φ with cos φ = y/s, giving E ∝ y/(R^2 + y^2)^(3/2)
   D. dE tan φ
   Answer: C. Perpendicular components cancel; the axial part has cos φ = y/s. (Key point: Axial component)

2. Why is the force on the electron toward the centre?
   A. The field points toward the ring for λ > 0
   B. The field points away from a positively charged ring but the electron's charge is negative
   C. The electron is attracted by gravity
   D. The ring has no field on the axis
   Answer: B. F = -eE reverses the direction of E. (Key point: Force direction)

3. For y << R, what is the force on the electron?
   A. F = -(eλ/2ε0 R^2) y, proportional to the displacement and opposite to it
   B. F = -(eλ/2ε0 R^2) y^2
   C. F = constant, toward the centre
   D. F = +(eλ/2ε0 R^2) y
   Answer: A. (R^2 + y^2)^(3/2) ≈ R^3 gives E ≈ λy/(2ε0 R^2). (Key point: Small y)

4. Which is the period of the electron's SHM?
   A. T = 2π sqrt(eλ/(2ε0 m R^2))
   B. T = sqrt(2ε0 m R^2/(eλ))
   C. T = 2π sqrt(ε0 m R/(eλ))
   D. T = 2π sqrt(2ε0 m R^2/(eλ))
   Answer: D. ω^2 = eλ/(2ε0 m R^2) and T = 2π/ω. (Key point: Period)

## Related in Engineering Physics

- [Field on axis of charged ring](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-axial-field-maximum) — Engineering Physics, chapter 7, asked 4×
- [Field at center of bent charged rod](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aplastic-rod-arc-field-center) — Engineering Physics, chapter 7
- [Ring on axis: potential then field](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-axial-potential-then-field) — Engineering Physics, chapter 7, asked 2×
- [Field inside uniformly charged cylinder](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acylinder-uniform-charge-field-inside) — Engineering Physics, chapter 7
- [Field on axis of charged disc](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adisc-axial-field) — Engineering Physics, chapter 7, asked 3×
- [Field of charged rod along its axis](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acharged-rod-field-along-axis) — Engineering Physics, chapter 7

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