Electric fields

Electric Fields – Damped and Forced Oscillations, Resonance

Learning Objective

To understand how damping and an external driving force modify simple harmonic motion (SHM), to describe the resulting resonance behaviour, to relate these mechanical ideas to the electric field in an oscillatory (RLC) circuit, and to recall the basic properties of electric fields and capacitance required by the Cambridge AS & A‑Level Physics syllabus.

Cambridge Syllabus Checklist (17.3 & 18‑19)

  • Define damped and forced oscillations; give sketches of under‑damped, critically damped and over‑damped motion.

  • Write the differential equation for a damped, driven mass‑spring system and identify each term.

  • State the resonance‑frequency shift for a damped system and derive ωr=√(ω₀²–γ²/2).

  • Define the quality factor Q and explain its effect on the resonance curve (peak height, bandwidth Δω=ω₀/Q).

  • Explain the phase relationship between driving force and displacement (or charge) and sketch the phase‑lag versus driving frequency.

  • Discuss energy loss per cycle and how it relates to the damping constant.

  • Define the electric field (E = F/q), draw field‑line patterns for a point charge and a parallel‑plate capacitor, and write the expression for a uniform field (E = ΔV/Δd).

  • State the definition of capacitance (C = Q/V), give the series and parallel combination formulas, and write the energy stored (U = ½CV²).

  • Identify the electrical analogue of a damped, driven mechanical oscillator (RLC series circuit) and relate the mechanical amplitude to the electric‑field amplitude in a capacitor.

Terminology (Cambridge wording)

TermMeaning (mechanical analogue)
Light dampingUnderdamped – the system oscillates while the amplitude decays exponentially.
Critical dampingBorderline case – returns to equilibrium as quickly as possible without overshooting.
Heavy dampingOver‑damped – returns slowly and does not oscillate.

Key Concepts

  • Simple harmonic motion (ideal, undamped).

  • Resistive (damping) force – always opposite to the instantaneous velocity and removes energy each cycle.

  • Light, critical and heavy damping and their effect on the motion.

  • Forced (driven) oscillations – a sinusoidal external force.

  • Resonance – maximum steady‑state amplitude when the driving frequency is close to the natural frequency.

  • Quality factor Q – a measure of how sharply the system resonates; bandwidth Δω = ω₀/Q.

  • Electrical analogue – an RLC series circuit obeys the same differential equation as a damped, driven mechanical oscillator.

  • Electric field – force per unit positive charge; uniform field between parallel‑plate capacitor plates.

  • Capacitance – ability of a system to store charge per unit potential difference.

1. Damped Oscillations

The equation of motion for a mass–spring system with a linear resistive force is

\$m\ddot{x}+b\dot{x}+kx=0\$

where

  • m – mass (kg)

  • b – damping coefficient (kg s⁻¹); the damping force is Fd=‑b v.

  • k – spring constant (N m⁻¹)

The discriminant \(\Delta=b^{2}-4mk\) distinguishes three regimes:

  1. Light (underdamped) damping (\(b^{2}<4mk\)) – oscillatory motion with envelope \(A(t)=A_{0}e^{-\gamma t}\) where \(\gamma=b/(2m)\).

  2. Critical damping (\(b^{2}=4mk\)) – fastest return to equilibrium without overshoot.

  3. Heavy (over‑damped) damping (\(b^{2}>4mk\)) – non‑oscillatory, slower return.

Solution forms

Define the natural angular frequency \(\omega_{0}=\sqrt{k/m}\) and the decay constant \(\gamma=b/(2m)\).

  • Underdamped: \(\displaystyle x(t)=A{0}e^{-\gamma t}\cos(\omega{d}t+\phi),\quad \omega{d}=\sqrt{\omega{0}^{2}-\gamma^{2}}\).

  • Critically damped: \(\displaystyle x(t)=(A{1}+A{2}t)e^{-\omega_{0}t}\).

  • Over‑damped: \(\displaystyle x(t)=A{1}e^{-(\gamma-\sqrt{\gamma^{2}-\omega{0}^{2}})t}+A{2}e^{-(\gamma+\sqrt{\gamma^{2}-\omega{0}^{2}})t}\).

2. Forced (Driven) Oscillations

With a sinusoidal driving force \(F{\rm d}(t)=F{0}\cos(\omega_{\rm d}t)\) the equation becomes

\$m\ddot{x}+b\dot{x}+kx = F{0}\cos(\omega{\rm d}t)\$

The steady‑state (particular) solution is

\$x(t)=A(\omega{\rm d})\cos\!\big(\omega{\rm d}t-\phi\big)\$

with

\[

A(\omega{\rm d})=\frac{F{0}/m}{\sqrt{(\omega{0}^{2}-\omega{\rm d}^{2})^{2}+(\gamma\omega_{\rm d})^{2}}},\qquad

\tan\phi=\frac{\gamma\omega{\rm d}}{\;\omega{0}^{2}-\omega_{\rm d}^{2}\;},

\]

where \(\gamma=b/m\) (note the factor‑2 difference from the envelope decay constant used above).

3. Resonance

Resonance occurs when the driving frequency maximises the amplitude. Differentiating \(A(\omega_{\rm d})\) gives the resonance angular frequency

\[

\boxed{\;\omega{\rm r}= \sqrt{\;\omega{0}^{2}-\dfrac{\gamma^{2}}{2}\;}\;}

\]

For light damping (\(\gamma\ll\omega{0}\)), \(\omega{\rm r}\approx\omega_{0}\). At exact resonance the phase lag is \(\phi=90^{\circ}\) (or \(\pi/2\) rad).

Quality factor and bandwidth

\[

Q=\frac{\omega_{0}}{\gamma}=\frac{1}{b}\sqrt{\frac{k}{m}},\qquad

\Delta\omega=\frac{\omega_{0}}{Q}

\]

A high Q → narrow resonance peak, large amplitude for a given driving force; a low Q → broad peak.

Resonance‑curve sketch (required by the syllabus)

Include a labelled graph of amplitude \(A\) (or electric‑field amplitude) versus driving angular frequency \(\omega_{\rm d}\). Mark:

  • Peak at \(\omega_{\rm r}\).

  • Half‑maximum points at \(\omega_{\rm r}\pm\Delta\omega/2\).

  • Bandwidth \(\Delta\omega\) and its relation to \(Q\).

  • Phase‑lag curve (0° → 180°) superimposed or shown alongside.

4. Electrical Analogue – RLC Series Circuit

The RLC series circuit obeys the same differential equation as a damped, driven mechanical oscillator:

\$L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{q}{C}= \mathcal{E}{0}\cos(\omega{\rm d}t)\$

Mechanical QuantityElectrical Analogue
Mass \(m\)Inductance \(L\)
Damping coefficient \(b\)Resistance \(R\)
Spring constant \(k\)Reciprocal of capacitance \(1/C\)
Displacement \(x\)Charge \(q\)
Force \(F\)EMF \(\mathcal{E}\)

Voltage across the capacitor:

\$V_{C}=\frac{q}{C}\qquad\Longleftrightarrow\qquad x\$

Electric field between the parallel‑plate capacitor (plate separation \(d\)):

\$E=\frac{V_{C}}{d}\$

Thus the amplitude of the mechanical oscillation corresponds directly to the amplitude of the electric field in the capacitor.

5. Electric Field Fundamentals (Syllabus 18)

  • Definition: \(\displaystyle \mathbf{E}=\frac{\mathbf{F}}{q}\) – the force \(\mathbf{F}\) experienced by a positive test charge \(q\) placed at a point.

  • Field‑line representation: lines start on positive charges and end on negative charges; density of lines ∝ field strength.

  • Uniform electric field: between large parallel plates,

    \[

    E=\frac{\Delta V}{\Delta d}

    \]

    where \(\Delta V\) is the potential difference and \(\Delta d\) the plate separation.

  • Force on a charge: \(\displaystyle \mathbf{F}=q\mathbf{E}\).

  • Potential energy change: \(\Delta U = -q\int\mathbf{E}\cdot d\mathbf{s}\).

6. Capacitance (Syllabus 19)

  • Definition: \(\displaystyle C=\frac{Q}{V}\) – the charge \(Q\) stored per unit potential difference \(V\).

  • Parallel‑plate capacitor: \(C=\varepsilon_{0}\frac{A}{d}\) (for vacuum), where \(A\) is plate area and \(d\) the separation.

  • Series combination: \(\displaystyle \frac{1}{C{\text{eq}}}= \sum{i}\frac{1}{C_{i}}\).

  • Parallel combination: \(\displaystyle C{\text{eq}}= \sum{i}C_{i}\).

  • Energy stored: \(\displaystyle U=\frac{1}{2}CV^{2}= \frac{Q^{2}}{2C}= \frac{1}{2}Q V\).

7. Common Misconceptions (Cambridge focus)

  • Resonance‑frequency shift: In heavily damped systems the resonance frequency is lower than the natural frequency; the correct expression is \(\omega{r}= \sqrt{\omega{0}^{2}-\gamma^{2}/2}\).

  • Amplitude at resonance: It depends on the driving amplitude and the quality factor; there is no single “maximum possible amplitude”.

  • Phase relationship: At resonance the displacement (or charge) lags the driving force by exactly \(90^{\circ}\); away from resonance the lag is <\(90^{\circ}\) for light damping and >\(90^{\circ}\) for heavy damping.

  • Effect of \(Q\): A larger \(Q\) narrows the resonance curve (smaller bandwidth) but does not automatically increase the absolute amplitude unless the driving force is held constant.

  • Damping force direction: The resistive force always opposes the instantaneous velocity, removing kinetic energy each cycle.

  • Electric‑field definition: Some students forget the “per unit positive charge” wording and treat the field as a vector independent of test charge sign.

8. Worked Example (Mechanical System)

Problem: A mass \(m=0.5\;\text{kg}\) is attached to a spring with \(k=200\;\text{N m}^{-1}\). The damping force is \(F{\rm d}=b\dot{x}\) with \(b=2\;\text{kg s}^{-1}\). The system is driven by a force \(F{0}=10\;\text{N}\) at a frequency \(f_{\rm d}=2\;\text{Hz}\). Find the steady‑state amplitude and the phase lag.

  1. Natural angular frequency: \(\displaystyle \omega_{0}= \sqrt{\frac{k}{m}}= \sqrt{\frac{200}{0.5}}=20\;\text{rad s}^{-1}\).

  2. Damping constant: \(\displaystyle \gamma=\frac{b}{m}= \frac{2}{0.5}=4\;\text{s}^{-1}\).

  3. Driving angular frequency: \(\displaystyle \omega{\rm d}=2\pi f{\rm d}=2\pi(2)=4\pi\;\text{rad s}^{-1}\approx12.57\;\text{rad s}^{-1}\).

  4. Amplitude:

    \[

    A=\frac{F{0}/m}{\sqrt{(\omega{0}^{2}-\omega{\rm d}^{2})^{2}+(\gamma\omega{\rm d})^{2}}}

    =\frac{10/0.5}{\sqrt{(400-158)^{2}+(4\times12.57)^{2}}}

    \approx5.8\times10^{-2}\;\text{m}=5.8\;\text{cm}.

    \]

  5. Phase lag:

    \[

    \tan\phi=\frac{\gamma\omega{\rm d}}{\omega{0}^{2}-\omega_{\rm d}^{2}}

    =\frac{4\times12.57}{400-158}\approx0.21\;\Longrightarrow\;\phi\approx12^{\circ}.

    \]

Hence the mass oscillates with an amplitude of about 5.8 cm and lags the driving force by ≈12°**.

9. Suggested Diagrams (for the teacher’s reference)

  • Sketch of a damped driven oscillator (mass‑spring‑damper) together with a side‑by‑side RLC series circuit. Arrows indicate the driving force \(F{0}\cos(\omega{\rm d}t)\) and the source EMF \(\mathcal{E}{0}\cos(\omega{\rm d}t)\).

  • Exponential‑decay envelope for the under‑damped case.

  • Resonance curve (amplitude vs. \(\omega{\rm d}\)) showing peak, half‑maximum points, bandwidth \(\Delta\omega\) and the relation \(\Delta\omega=\omega{0}/Q\).

  • Phase‑lag versus driving frequency plot (0° → 180°).

  • Field‑line diagram for a point charge and for a parallel‑plate capacitor (uniform field). Include the relation \(E=\Delta V/d\).

  • Capacitance combination diagram (series and parallel) with formulas highlighted.