GTU Diploma Engineering Physics - Comprehensive Examination Winter 2025 Study Guide

Measurements, Units, and Instrumentation in Physics

• Significant Figures: The concept of significant figures is used to indicate the precision of a measurement. It includes all certain digits plus one final uncertain digit.

• Rules for Counting Significant Figures:

  • All non-zero digits are significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros (to the left of the first non-zero digit) are not significant; they only indicate decimal placement.
  • Trailing zeros after a decimal point are significant.

• Application to Provided Values:

  • (i) 48956: There are 5 significant figures (all are non-zero digits).
  • (ii) 1.0023: There are 5 significant figures (zeros between non-zero digits are counted).
  • (iii) $0.02 \times 10^{-31}$: There is 1 significant figure (leading zeros "0.0" are not counted, and the exponent $10^{-31}$ does not affect significant figure count).

• Vernier Caliper Construction and Working:

  • Main Scale: A fixed scale with markings in mm or cm.
  • Vernier Scale: A sliding scale that allows for more precise readings than the main scale can provide alone.
  • Jaws: External jaws for measuring outside diameters/thickness and internal jaws for measuring inner diameters.
  • Depth Rod/Stem: Used to measure the depth of holes or cavities.
  • Least Count (LC): The smallest measurement that can be made. It is calculated as: $LC = 1\text{ Main Scale Division (MSD)} - 1\text{ Vernier Scale Division (VSD)}$.
  • Total Reading: The final value is calculated as: $\text{Total Reading} = \text{Main Scale Reading (MSR)} + (\text{Vernier Scale Reading (VSR)} \times LC)$.

• Error Analysis in Measurements (Ohm’s Law Example):

  • Measured Values (Resistance): $R_1 = 2.00\,\Omega$, $R_2 = 2.10\,\Omega$, $R_3 = 2.15\,\Omega$, $R_4 = 2.05\,\Omega$.
  • Mean Value (True Value): $R_{mean} = \frac{2.00 + 2.10 + 2.15 + 2.05}{4} = 2.075\,\Omega$.
  • Absolute Errors: $|\Delta R_1| = 0.075\,\Omega$, $|\Delta R_2| = 0.025\,\Omega$, $|\Delta R_3| = 0.075\,\Omega$, $|\Delta R_4| = 0.025\,\Omega$.
  • Mean Absolute Error: $\Delta R_{mean} = \frac{0.075 + 0.025 + 0.075 + 0.025}{4} = 0.05\,\Omega$.
  • Percentage Error: $\text{Percentage Error} = (\frac{\Delta R_{mean}}{R_{mean}}) \times 100 = (\frac{0.05}{2.075}) \times 100 \approx 2.41\%$.

Electrostatics and Capacitance

• Dielectric Materials: These are insulating materials that do not conduct electricity but can support an electrostatic field. Examples include Glass, Mica, Plastic, and Ceramic.
• Coulomb’s Law: It states that the force of attraction or repulsion between two point charges ($q_1$ and $q_2$) is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance ($r$) between them.
  • Expression: $F = k \cdot \frac{q_1 \cdot q_2}{r^2}$. Where $k \approx 9 \times 10^9\,N\,m^2/C^2$ in a vacuum.
• Calculated Electrostatic Force: For charges $4\,\mu c$ ($4 \times 10^{-6}\,C$) and $6\,\mu c$ ($6 \times 10^{-6}\,C$) separated by $2\,m$:
  • $F = (9 \times 10^9) \cdot \frac{(4 \times 10^{-6}) \cdot (6 \times 10^{-6})}{2^2} = \frac{216 \times 10^{-3}}{4} = 0.054\,N$.
• Electric Field Lines Characteristics:
  • They originate from positive charges and terminate at negative charges.
  • They never intersect each other.
  • The tangent to a field line at any point gives the direction of the electric field at that point.
• Parallel Plate Capacitor: Consists of two parallel conducting plates separated by a dielectric.
  • Expression Derivation: For plates of area $A$ and separation $d$, with surface charge density $\sigma = Q/A$, the electric field $E = \frac{\sigma}{\epsilon_0}$. Since potential $V = E \cdot d$, we get $V = \frac{Q \cdot d}{A \cdot \epsilon_0}$. Capacitance $C = \frac{Q}{V}$, leading to: $C = \frac{\epsilon_0 \cdot A}{d}$.

Thermodynamics and Heat Measurement

• Temperature Scales Interrelation: Temperature can be measured in Celsius ($^{\circ}C$), Fahrenheit ($^{\circ}F$), and Kelvin ($K$).
  • Interrelation Formula: $\frac{C}{5} = \frac{F - 32}{9} = \frac{K - 273.15}{5}$.
• Temperature Conversion Example (200 Kelvin):
  • To Celsius: $C = K - 273.15 = 200 - 273.15 = -73.15^{\circ}C$.
  • To Fahrenheit: $F = (\frac{9}{5} \cdot C) + 32 = (\frac{9}{5} \cdot -73.15) + 32 \approx -99.67^{\circ}F$.
• Mercury Thermometer:
  • Principle: Thermal expansion of liquids (mercury expands uniformly with heat).
  • Advantage: Mercury remains liquid over a wide range of temperatures and does not wet the glass.
  • Disadvantage: It is highly toxic if broken and has a relatively low freezing point compared to some industrial needs.
  • Application: Used in clinical and laboratory temperature measurements.
• Linear Thermal Expansion: The change in length ($\Delta L$) due to temperature change ($\Delta T$) is given by $\Delta L = L_0 \cdot \alpha \cdot \Delta T$.
  • Problem Calculation: For a rod of $1\,m$ at $0^{\circ}C$ heated to $90^{\circ}C$:
  • Steel: $\alpha_s = 11 \times 10^{-6}\,{^{\circ}C^{-1}}$. $\Delta L_s = 1 \cdot 11 \times 10^{-6} \cdot 90 = 990 \times 10^{-6}\,m$.
  • Aluminium: $\alpha_a = 24 \times 10^{-6}\,{^{\circ}C^{-1}}$. $\Delta L_a = 1 \cdot 24 \times 10^{-6} \cdot 90 = 2160 \times 10^{-6}\,m$.
  • Difference: $|2160 - 990| \times 10^{-6} = 1170 \times 10^{-6}\,m$ or $1.17\,mm$.
• Heat Capacity: The amount of heat required to raise the temperature of a substance by one Kelvin or one degree Celsius. SI unit: $J\,K^{-1}$.

Waves, Sound, and Ultrasonics

• Fundamental Wave Definitions:
  • Wavelength (\lambda): The distance between two consecutive peaks or troughs in a wave.
  • Amplitude (A): The maximum displacement of a particle from its mean position.
• Sound Wave Characteristics:
  • Requires a material medium for propagation (longitudinal waves).
  • Characterized by frequency, amplitude, and speed.
• Acoustics of Buildings:
  • Reverberation Time: The time taken for the sound to drop by $60\,dB$ after the source stops.
  • Sabine’s Formula: $T = \frac{0.161 \cdot V}{A}$, where $V$ is volume and $A$ is total absorption.
  • Absorption Calculation: For $V = 3000\,m^3$ and $T = 1.7\,s$: $1.7 = \frac{0.161 \times 3000}{A} \implies A = \frac{483}{1.7} \approx 284.12\,\text{Sabines}$.
• Ultrasonic Waves: Sound waves with frequencies higher than the human audible limit (greater than $20\,kHz$).
  • Characteristics: Highly energetic, small wavelength (high resolution), and can travel long distances without much spreading.
• Piezoelectric Generator: Based on the Piezoelectric effect, where mechanical pressure on certain crystals (like quartz) generates an electric potential.
  • Working: Alternating potential is applied to the crystal edges to produce ultrasonic vibrations via resonance.
• Interference of Sound: The phenomenon where two waves superimpose to form a resultant wave of greater, lower, or the same amplitude.
  • Constructive: Waves meet in phase, increasing amplitude.
  • Destructive: Waves meet out of phase, decreasing amplitude.
• Wave Speed Calculation: Given frequency $f = 250\,Hz$ and velocity $v = 330\,m/s$:
  • $\lambda = \frac{v}{f} = \frac{330}{250} = 1.32\,m$.

Optics and Laser Physics

• LASER (Light Amplification by Stimulated Emission of Radiation):
  • Properties: Monochromatic (single wavelength), Coherent (waves are in phase), Highly Directional/Collimated, and High Intensity.
  • Advantages over Ordinary Light: Laser light can be focused to very small spots, travels over long distances without divergence, and carries high energy for cutting or surgical applications.
• Laws of Reflection:
  • The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.
  • The angle of incidence ($\theta_i$) is equal to the angle of reflection ($\theta_r$).
• Optical Fiber Structure:
  • Core: The inner thin glass or plastic through which light travels.
  • Cladding: The outer layer with a lower refractive index that reflects light back into the core via Total Internal Reflection (TIR).
  • Buffer Coating: Protective layer for the fiber.
• Fiber Optics Math (Example 1: Core $n_1 = 1.5$, Cladding $n_2 = 1.4$):
  • Numerical Aperture (NA): $NA = \sqrt{n_1^2 - n_2^2} = \sqrt{1.5^2 - 1.4^2} = \sqrt{2.25 - 1.96} = \sqrt{0.29} \approx 0.5385$.
  • Acceptance Angle (\theta_a): $\theta_a = \sin^{-1}(NA) = \sin^{-1}(0.5385) \approx 32.58^{\circ}$.
  • Critical Angle (\theta_c): $\theta_c = \sin^{-1}(\frac{n_2}{n_1}) = \sin^{-1}(\frac{1.4}{1.5}) \approx 68.96^{\circ}$.
• Fiber Optics Math (Example 2: Core $n_1 = 1.48$, Relative Index $\Delta = 2\% = 0.02$):
  • Since $\Delta = \frac{n_1 - n_2}{n_1}$, then $n_2 = n_1(1 - \Delta) = 1.48(1 - 0.02) = 1.4504$.
  • Critical Angle: $\sin^{-1}(\frac{1.4504}{1.48}) = \sin^{-1}(0.98) \approx 78.52^{\circ}$.
• Applications of Optical Fiber:
  • Engineering: High-speed telecommunication, data transmission, and sensors for detecting structural stress.
  • Medical: Endoscopy (visualizing internal organs) and laser surgery (delivering precise light energy).