---
title: "Sharpness of resonance, forced e-m oscillation"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:forced-em-sharpness-current-amplitude"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-em-sharpness-current-amplitude
site: BCT Engineering
---

# Sharpness of resonance, forced e-m oscillation

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2073 Shrawan, 5 marks
- Video: [Forced Vibration: Amplitude and Velocity, Resonance, Sharpness of Resonance, LCR Circuit](https://www.youtube.com/watch?v=4ZK0s5RNDOA) — Mywbut
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/forced-em-sharpness-current-amplitude
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

Define sharpness of resonance. Derive the relation for current amplitude of forced e-m oscillation.

## Short answer

Sharpness describes resonance peak width; Q = ω₀L/R. For a series LCR circuit with E = E₀ cos ωt, current amplitude is I₀ = E₀ / √[R² + (ωL − 1/ωC)²]. Maximum I₀ = E₀/R occurs at ω = 1/√(LC). Smaller R gives sharper resonance.

## Answer

#### Sharpness of resonance
It describes how rapidly the current amplitude (or any response) drops when the driving frequency moves away from the resonant frequency. Small damping (small R) gives a sharp resonance. It is measured by the quality factor Q = ω0L/R = ω0/Δω, where Δω is the half-power bandwidth.

#### Current amplitude of forced e.m. oscillation
A series L, C, R circuit with applied emf E = E0 cos ωt.
```text
 ~E0 cos ωt ---- L ---- R ---- C ---+
   |_________________________________|
```
1. Equation: L d²q/dt² + R dq/dt + q/C = E0 cos ωt.
2. In the steady state the charge oscillates at the driving frequency: q = q0 cos(ωt − φ).
3. Substitute, write cos ωt = cos(ωt − φ)cosφ − sin(ωt − φ)sinφ, and equate coefficients:
```text
q0 (1/C − Lω²) = E0 cosφ
q0 R ω         = E0 sinφ
```
4. Square and add:
```text
q0 = E0 / {ω √[R² + (ωL − 1/ωC)²]}
```
5. The current is i = dq/dt = −ωq0 sin(ωt − φ), so the current amplitude is I0 = ωq0:
```text
I0 = E0 / √[R² + (ωL − 1/ωC)²] = E0/Z
```
The current lags the emf by ψ = φ − π/2, where tanψ = (ωL − 1/ωC)/R.
6. I0 is maximum (E0/R) when ωL = 1/ωC, i.e. ω = 1/√(LC). The smaller R is, the taller and sharper the peak of I0 against ω.

## Check yourself

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

1. Sharpness of resonance is?
   A. The maximum current
   B. The resonant frequency
   C. The source emf
   D. How rapidly response drops away from resonance
   Answer: D. Measured by Q. (Key point: Sharpness)

2. In steady state charge oscillates at?
   A. The driving frequency
   B. w0
   C. w'
   D. Zero
   Answer: A. Transient dies out. (Key point: Steady state)

3. I0 =?
   A. E0 R
   B. E0 / sqrt(R^2 + (wL - 1/wC)^2)
   C. E0 / (wL)
   D. E0 sqrt(R^2 + X^2)
   Answer: B. I0 = E0/Z. (Key point: Current amplitude)

4. I0 maximum is?
   A. E0/Z at all w
   B. E0 at w = 0
   C. E0/R at w = 1/sqrt(LC)
   D. E0 L/R
   Answer: C. X = 0 at resonance. (Key point: Maximum)

## Related in Engineering Physics

- [Forced vibration amplitude and resonance](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-oscillation-amplitude-resonance) — Engineering Physics, chapter 1, asked 2×
- [Forced LCR oscillation and resonance](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-resonance-frequency) — Engineering Physics, chapter 1, asked 2×
- [Sharpness of resonance and quality factor](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Asharpness-resonance-quality-factor) — Engineering Physics, chapter 1
- [Damped vs forced LCR oscillation](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aforced-lcr-differential-compare) — Engineering Physics, chapter 1

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