CBSE Class 11 - Physics - CH2 - Error Analysis #PhysicsNotes #eduvictors

Error Analysis

Q1: Define error.

Answer: The exact measurement of a physical quaintly is not possible. This is known as Heisenberg's uncertainly principle. Uncertainty in the measurements is called error.

Numerically, the deviation of the measured value from the actual value is called an error.  If X is the quantity measured, then ΔX is called error in X. i.e. Error = Actual value - Measured value.




Q2: What are different sources of errors?

Answer: An error can be of the following types:
(a) systematic errors
(b) random errors.
(c) least count errors
(d) constant errors
(e) Errors due to external factors.

(a) Systematic error is the deviation caused due a single known source is called a systematic error. Following are the sources of systematic errors:

• Instrumental errors: Imperfect design or calibration and/or Zero error.
• Imperfection in experimental technique or procedure
• Personal errors or Gross errors due to individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions,

(b) Random Error: Error due to unknown reasons and that occur while taking measurements.

(c) The least count error is the error associated with the resolution of the instrument.

(d) Constant Errors. Error in magnitude caused due to problems in the calibration of scales. This error repeats itself in all measurements. The least Count error is also a type of constant error.

(e) External factors like air, temperature, pressure etc. can cause errors while taking measurements.

Q3: How can we minimize errors?

Answer:

1. Using instruments of higher precision, improving experimental techniques, etc., we can
   reduce the least count error
2. Repeating the observations several times and taking the arithmetic mean of all the observations, the mean value would be very close to the true value of the measured quantity.
3. Gross errors can be minimized only if the observer is very careful in his observations and sincere in his approach.

Q4: What are the different ways of expressing an error?

Answer: different ways to express errors are:

• Absolute Error
• Relative Error
• Percentage Error

Q5: What is Absolute Error?

Answer:  The absolute error of measurement is the magnitude of the difference between the value of the quantity and the individual measurement value. Because we are not sure about the correct value of the quantity, the correct or true value of a quantity is taken as the Arithmetic mean (A_mean) of reading values.

∴  If there are 'n' number of readings (A₁, A₂, A₃ ... A_n), corresponding absolute errors can be represented as:
   ΔA₁ = A_mean - A₁
   ΔA₂ = A_mean - A₂
   ΔA₃ = A_mean - A₃
...
   ΔA_n = A_mean - A_n

Final Absolute error is the arithmetic mean of magnitudes of absolute errors i.e.
ΔA_mean= ( |ΔA₁|+ |ΔA₂| + |ΔA₃|+ ... + |ΔA_n|) / n

Q6: Define accuracy.

Answer: The accuracy of an instrument is a measure of how close the output reading of the instrument is to the correct value.

Q7: What is meant by precision? Can we say an instrument of high precision is accurate?


Answer: Precision describes an instrument’s degree of freedom from random errors. If a large number of readings are taken of the same quantity by a high-precision instrument, then the spread of readings will be very small. It is not necessarily for an instrument of high precision to be accurate.

Q8: Any measurement of physical quantity p will likely have a value
(a) Always equal to p_mean - Δp_mean
(b) Always equal to p_mean + Δp_mean
(c) Greater than  p_mean + Δp_mean
(d) Lies between  p_mean - Δp_mean ≤  p  ≤  p_mean + Δp_mean

Answer:  (d) Lies between  p_mean - Δp_mean ≤  p  ≤  p_mean + Δp_mean

Q9: What is a relative error or percentage error?

Answer: The relative error is the ratio of the mean absolute error Δa_mean to the mean value a_mean of the quantity measured.
i.e. Relative error = Δa_mean/ a_mean

When the relative error is expressed in per cent, it is called the percentage error (δa).
⇒  δa = (Δa_mean/ a_mean) × 100%

Q10: A resistor has a marking as “470 Ω, 10%”? What will be the true value of the resistor?

Answer: 470 Ω, 10% means, the resistor has a percentage error of 10%. it means the true value lies within 470 ± 10% i.e. 470 ± 47. (10% of 470 is 47)
∴ the true value lies between 423Ω to 517 Ω.

Q11: The refractive index (μ) of water is found to have the values 1.29, 1.33, 1.34, 1.35, 1.32, 1.36, 1.30 and 1.33. Calculate the mean value, absolute error, relative error and percentage error. 

Answer: Mean refractive index is an average of eight values i.e.
  μ_mean = (1.29 + 1.33 + 1.34 + 1.35 + 1.32 + 1.36 + 1.30 + 1.33) ÷ 8
  μ_mean =10.62 ÷ 8 = 1.3275 ≈ 1.33

Absolute error are:
Δμ₁ = μ_mean - μ₁  = 1.33 - 1.29 = 0.04
Δμ₂ = μ_mean - μ₂  = 1.33 - 1.33 = 0.00
Δμ₃ = μ_mean - μ₃  = 1.33 - 1.34 = -0.01
Δμ₄ = μ_mean - μ₄  = 1.33 - 1.35 = -0.02
Δμ₅ = μ_mean - μ₅  = 1.33 - 1.32 = 0.01
Δμ₆ = μ_mean - μ₆  =  1.33 - 1.36 = -0.03
Δμ₇ = μ_mean - μ₇  = 1.33 - 1.30 = 0.03
Δμ₈ = μ_mean - μ₈  =  1.33 - 1.33 = 0.00

Mean Absolute error  Δμ_mean
        = (|Δμ₁ | + |Δμ₂ | + |Δμ₃ | + | Δμ₄ | + | Δμ₅ | + | Δμ₆ | + |Δμ₇ | + |Δμ₈ |) ÷ 8
        = ( 0.04 + 0.00 + 0.01 + 0.02 + 0.01 + 0.03 + 0.03 + 0.00) ÷ 8
        = 0.14 ÷ 8 = 0.0175 ≈  0.02

Relative error (δμ) =  Δμ_mean/ μ_mean = 0.02 ÷ 1.33 = ± 0.015 ≈ 0.02

Percentage error =  (Δμ_mean/ μ_mean) × 100% = ± 0.015 × 100 = ± 1.5

Q12: Static analysis is used to handle
(a) Systematic errors
(b) Random errors
(c) Constant errors
(d) Least Count errors

Answer: (b) Random errors

Q13: What is a Parallax error?

Answer:  Parallax error is a type of gross error which gets introduced by reading scales from the wrong angle i.e. any angle other than at a right angle. Usually, some instruments have mirrors that help to avoid parallax errors while taking readings.

Q14: How errors are propagated or combined?

Answer: While applying mathematical operations (e.g. addition, subtraction, multiplication and division) on physical quantities, errors are combined.

If A and B are two physical quantities represented as A ± ΔA and B ± ΔB, then
(a) In addition, i.e. Z = A + B, absolute error  ΔZ = ΔA + ΔB.

(b) In difference, Z = A - B, absolute error  ΔZ = ΔA + ΔB.

(c) In product, Z = AB, then absolute error  ΔZ/Z = ΔA/A + ΔB/B.

(d) In division, Z = A/B,  ΔZ/Z = ΔA/A + ΔB/B.

(e) In power, Z = A^x.B^y/C^z, then  ΔZ/Z = xΔA/A + yΔB/B ± zΔC/C


Q15: A current of 3.5 ± 0·5 ampere flows through a metallic conductor.  And a potential difference of 21 ± 1 volt is applied. Find the Resistance of the wire.


Answer:  Given V = 21± 1volts, ΔV = 1, I = 3.5 ± 0·5 A and ΔI = 0.5A
Resistance R  = V / I = 21± 1 / 3.5 ± 0·5 = 6 ± ΔR
ΔR/R = error in measurement = ΔV/V + ΔI/I = 1/21 + 0.5/3.5 = 0.048 + 0.143 = 0.191 ≈ 0.19
⇒ ΔR = 0.19 × R = 0.19 × 6 = 1.14

∴ effective resistance R = 6 ± 1.14 Ω

Q16: A rectangular board is measured with a scale having an accuracy of 0.2cm. The length and breadth are measured as 35.4 cm and 18.4 cm respective. Find the relative error and percentage error of the area calculated.

Answer: Given length (l) = 35.4 cm, Δl = 0.2cm
Width (w) = 18.4cm and Δw = 0.2cm

Area (A) = l × w = 35.4 × 18.4 = 651.36 cm²

Relative error in Area (δA) = ΔA/A = Δl/l + Δw/w =0.2/35.4 + 0.2/18.4 = 0.006 + 0.011 = 0.017
Percentage error = ΔA/A × 100 = 0.017 × 100 = 1.7%

(In progress...)