understand that the resistance of a light-dependent resistor (LDR) decreases as the light intensity increases

1 Physical Quantities & Units

• Scalar vs vector – scalars have magnitude only (e.g. mass, temperature); vectors have magnitude and direction (e.g. displacement, force).
• SI base units – metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
• Derived units (selected):

  • Velocity : m s⁻¹
  • Acceleration : m s⁻²
  • Force : newton (N) = kg m s⁻²
  • Pressure : pascal (Pa) = N m⁻²
  • Energy : joule (J) = N m
  • Power : watt (W) = J s⁻¹
  • Resistance : ohm (Ω) = V A⁻¹

• Prefixes – milli (m = 10⁻³), centi (c = 10⁻²), kilo (k = 10³), mega (M = 10⁶), etc.
• Unit‑check checklist – before using a formula, confirm that the units on both sides are identical (e.g. J = N m = kg m² s⁻²).

2 Kinematics (uniform acceleration)

• Displacement, s – vector, SI unit m.
• Velocity, v – v = Δs/Δt, unit m s⁻¹.
• Acceleration, a – a = Δv/Δt, unit m s⁻².

Key equations (derived from constant‑a assumption):

\[

v = u + at,\qquad

s = ut + \tfrac12 a t^{2},\qquad

v^{2}=u^{2}+2as

\]

Example: A cart starts from rest (u = 0) and accelerates at 2 m s⁻² for 3 s.

\(v = 0 + 2\times3 = 6\; \text{m s}^{-1}\);

\(s = 0\times3 + \tfrac12\times2\times3^{2}=9\; \text{m}\).

3 Dynamics

• Newton’s First Law – an object remains at rest or in uniform motion unless acted on by a net external force.
• Newton’s Second Law – \(\displaystyle \mathbf{F}=m\mathbf{a}\). Direction of \(\mathbf{F}\) is the same as \(\mathbf{a}\).
• Newton’s Third Law – for every action there is an equal and opposite reaction.
• Momentum – \(\mathbf{p}=m\mathbf{v}\); conserved in isolated systems.
• Friction – kinetic friction \(F{k}= \mu{k}N\); static friction \(F{s}\le \mu{s}N\). \(\mu\) is dimensionless.

4 Forces, Density & Pressure

• Resultant force – vector sum of all individual forces.
• Weight – \(W = mg\) (g ≈ 9.81 m s⁻²).
• Density – \(\rho = \dfrac{m}{V}\) (kg m⁻³).
• Pressure – \(P = \dfrac{F}{A}\) (Pa). For fluids, hydrostatic pressure \(P = \rho gh\).

5 Work, Energy & Power

• Work – \(W = \mathbf{F}\cdot\mathbf{s}=Fs\cos\theta\) (J). Positive when force has a component along the displacement.
• Kinetic energy – \(K = \tfrac12 mv^{2}\).
• Gravitational potential energy – \(U = mgh\) (reference level chosen as convenient).
• Energy conservation – total mechanical energy \(E = K + U\) is constant in the absence of non‑conservative forces.
• Power – \(P = \dfrac{W}{t}=Fv = IV = I^{2}R = \dfrac{V^{2}}{R}\) (W).
• Efficiency – \(\displaystyle \eta = \frac{\text{useful output energy}}{\text{input energy}}\times100\%.\)

6 Deformation of Solids

• Stress – \(\sigma = \dfrac{F}{A}\) (Pa). Tensile stress when pulling, compressive when pushing.
• Strain – \(\varepsilon = \dfrac{\Delta L}{L_{0}}\) (dimensionless).
• Hooke’s Law – within the elastic limit, \(\sigma = E\varepsilon\) where \(E\) is Young’s modulus (Pa).
• Applications – strain‑gauge sensors, load cells, and the principle behind the change of resistance in some sensors.

7 Waves (basic concepts required for optics & sensors)

• Wave description – transverse vs longitudinal; characterised by wavelength \(\lambda\), frequency \(f\), speed \(v\) with \(v = f\lambda\).
• Superposition – principle of interference and standing waves (relevant for optical sensors).
• Reflection & refraction – Snell’s law \(n{1}\sin\theta{1}=n{2}\sin\theta{2}\); critical angle for total internal reflection.

8 Resistance, Resistivity & Conductivity

• Resistance \(R\) – \(R = \dfrac{V}{I}\) (Ω). Opposition to charge flow.
• Resistivity \(\rho\) – intrinsic property of a material (Ω·m). For a uniform conductor

  \[

  R = \rho\,\frac{L}{A}

  \]

  where \(L\) is length and \(A\) cross‑sectional area.

• Conductivity \(\sigma\) – reciprocal of resistivity, \(\sigma = \dfrac{1}{\rho}\) (S·m⁻¹).
• Temperature dependence (metals) –

  \[

  \rho(T)=\rho{0}\bigl[1+\alpha\,(T-T{0})\bigr]

  \]

  with \(\alpha>0\). For many semiconductors \(\alpha\) is negative, so resistivity falls as temperature rises.

• Power dissipation –

  \[

  P = VI = I^{2}R = \frac{V^{2}}{R}

  \]

  (W). Important when assessing heating of resistors or LDRs.

9 DC Circuits (Cambridge AS & A‑Level Physics 9702)

9.1 Practical circuits – symbols, EMF, internal resistance, safety

• Circuit symbols (Cambridge standard):

  • Battery (EMF source): ⎓
  • Resistor (including LDR): ⎓ (zig‑zag line)
  • Voltmeter: V (circle with “V”)
  • Ammeter: A (circle with “A”)

• EMF \(\mathcal{E}\) – energy supplied per coulomb. Terminal potential difference is \(\mathcal{E}-Ir{\text{int}}\) where \(r{\text{int}}\) is the battery’s internal resistance.
• Safety reminders:

  • Never exceed the rated voltage or current of a component.
  • Always switch off the supply before rewiring.
  • Use a current‑limiting resistor when testing an LDR with a fresh battery.
  • Check connections for secure contacts to avoid overheating.

9.2 Kirchhoff’s Laws – series and parallel combinations

• KCL (Current Law) – algebraic sum of currents entering a junction is zero.
• KVL (Voltage Law) – algebraic sum of potential differences round any closed loop is zero.
• Series – \(R{\text{eq}} = R{1}+R_{2}+…\); same current through each element, total voltage is the sum of individual drops.
• Parallel – \(\displaystyle \frac{1}{R{\text{eq}}}= \frac{1}{R{1}}+\frac{1}{R_{2}}+…\); same voltage across each branch, total current is the sum of branch currents.

9.3 Potential dividers – using an LDR (or thermistor) in a null method

A potential divider consists of two series resistances powered by a constant EMF. The output voltage at the junction is

\[

V{\text{out}} = \mathcal{E}\,\frac{R{2}}{R{1}+R{2}}

\]

When \(R{2}\) is an LDR, \(V{\text{out}}\) varies with light intensity. In a null‑method measurement a second, known resistor \(R_{\text{ref}}\) is adjusted until a galvanometer reads zero between the divider output and a calibrated reference voltage. At null:

\[

\frac{R{\text{LDR}}}{R{\text{ref}}}= \frac{V{\text{ref}}}{\mathcal{E}-V{\text{ref}}}

\]

Thus the LDR resistance is obtained directly without measuring current.

10 Light‑Dependent Resistor (LDR)

10.1 Construction and operation

• Active layer – thin film of a high‑resistivity semiconductor, most commonly cadmium sulfide (CdS), deposited on an insulating substrate.
• Electrical contacts are painted on opposite sides of the film.
• When photons with energy ≥ the band‑gap strike the film, electrons are excited to the conduction band, increasing the free‑carrier concentration. Consequently conductivity \(\sigma\) rises and resistivity \(\rho\) (hence resistance) falls.

10.2 Resistance versus illuminance

Empirically the relationship follows a power law:

\[

R = k\,I^{-\alpha}

\]

• \(I\) – illuminance in lux (lx).
• \(k\) – device‑specific constant (Ω·lx\(^{\alpha}\)).
• \(\alpha\) – light‑sensitivity exponent, typically \(0.5 \le \alpha \le 1.0\).

Thus, as the light level increases, the resistance decreases sharply.

10.3 Typical resistance values

10.4 Experimental verification

1. Series‑circuit method (Ohm’s law)

   • Connect the LDR in series with a known resistor \(R_{\text{ref}}\) and a DC source (e.g. 5 V).
   • Measure the voltage across the LDR, \(V_{\text{LDR}}\), for several lighting conditions.
   • Current in the loop: \(I = \dfrac{\mathcal{E}-V{\text{LDR}}}{R{\text{ref}}}\).
   • Calculate \(R{\text{LDR}} = \dfrac{V{\text{LDR}}}{I}\).

2. Potential‑divider (null) method

   • Form a voltage divider with \(R_{\text{ref}}\) and the LDR.
   • Adjust \(R_{\text{ref}}\) (or a potentiometer) until a galvanometer reads zero between the divider output and a calibrated reference voltage.
   • At null: \(\displaystyle \frac{R{\text{LDR}}}{R{\text{ref}}}= \frac{V{\text{ref}}}{\mathcal{E}-V{\text{ref}}}\), giving \(R_{\text{LDR}}\) directly.

3. Record the corresponding illuminance with a lux‑meter.
4. Plot \(\log(R_{\text{LDR}})\) against \(\log(I)\). The slope equals \(-\alpha\), confirming the power‑law behaviour.

10.5 Sample calculation (voltage divider)

Given \(R_{\text{ref}}=10\; \text{k}\Omega\), \(\mathcal{E}=5\; \text{V}\) and an illuminance of \(200\; \text{lx}\). Assume \(k=10^{5}\; \Omega\cdot\text{lx}^{\alpha}\) and \(\alpha=0.7\).

1. Resistance of the LDR:

   \[

   R_{\text{LDR}} = 10^{5}\,(200)^{-0.7}\approx 7.9\;\text{k}\Omega.

   \]

2. Output voltage:

   \[

   V_{\text{out}} = 5\;\frac{7.9}{10+7.9}\approx 2.9\;\text{V}.

   \]

3. If the light level is increased to \(2\,000\; \text{lx}\):

   \[

   R_{\text{LDR}}\approx 2.0\;\text{k}\Omega,\qquad

   V_{\text{out}}\approx 1.4\;\text{V},

   \]

   clearly showing the inverse relationship between light intensity and resistance.

11 Summary of Key Points

• Physical quantities are expressed in SI units; always check dimensional consistency.
• Kinematics and dynamics provide the foundation for understanding forces that may act on circuit components (e.g. moving‑coil meters).
• Work, energy and power link mechanical and electrical phenomena; power formulas \(P=IV=I^{2}R=V^{2}/R\) are used repeatedly.
• Stress, strain and Hooke’s law are relevant for sensors that change resistance when deformed.
• Waves concepts (frequency, wavelength, superposition) underpin optical sensors such as LDRs.
• Resistance depends on material resistivity, geometry and temperature; semiconductors (including CdS) show a negative temperature coefficient.
• DC‑circuit analysis (symbols, EMF, internal resistance, Kirchhoff’s laws, potential dividers) is essential for designing and interpreting LDR circuits.
• An LDR is a semiconductor photo‑resistor whose resistance follows \(R=kI^{-\alpha}\); resistance falls dramatically as illuminance rises.
• Typical LDR resistances range from \(\sim\)MΩ in darkness to a few hundred Ω in bright sunlight.
• Experimental determination of the \(R\)–\(I\) relationship can be carried out with a simple series circuit or a null‑method voltage divider; a log‑log plot yields the exponent \(\alpha\).