Given below are two statements: Statement (I): The dimensions of Planck's constant and angular momentum are same. Statement (II): In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant. In the light of the above statements, choose the most appropriate answer from the options given below:

Solution & Explanation

### Related Formula E = hf implies [h] = frac[E][f] = fractextMtextL^2textT^-2textT^-1 = textMtextL^2textT^-1 L = mvr implies [L] = textM cdot (textLtextT^-1) cdot textL = textMtextL^2textT^-1 L = fracnh2pi ### Core Logic Statement I: Comparing the dimensional formula of Planck's constant (h) and angular momentum (L), both are identical [textMtextL^2textT^-1]. Hence, Statement I is correct. Statement II: According to Bohr's second postulate, angular momentum is an integral multiple of frach2pi, not an integral multiple of h. Hence, Statement II is incorrect. ### Pattern Recognition Watch out for exact definitions in standard postulates. Bohr's model requires angular momentum to be quantized in units of hbar = frach2pi, making statement II a classic trap. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements Class 12 Physics: Atoms

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Q50 jee_main_2024_31_jan_evening Dimensional Analysis
Consider two physical quantities A and B related to each other as E = fracB - x^2At where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is
  • A. textL^-2textM^1textT^0
  • B. textL^2textM^-1textT^1
  • C. textL^-2textM^-1textT^1
  • D. textL^0textM^-1textT^1

Solution

### Related Formula By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x^2] ### Core Logic Known dimensional formulas: Length x to [L] Energy E to [ML^2T^-2] Time t to [T] ### Step 1: Dimension of B Since x^2 is subtracted from B: [B] = [x^2] = [L^2] ### Step 2: Dimension of A From the equation E = fracB - x^2At: [A] = frac[B - x^2][E][t] [A] = frac[L^2][ML^2T^-2][T] = frac[L^2][ML^2T^-1] [A] = [M^-1T^1] ### Step 3: Dimension of AB [AB] = [A] times [B] [AB] = [M^-1T^1] times [L^2] [AB] = [L^2 M^-1 T^1] ### Pattern Recognition Identify sums/differences first to instantly isolate B. Once [B] is fixed, the entire numerator is just L^2. Swap out variables to isolate [A]. Combining is just standard exponent addition. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units and Measurements
Q39 jee_main_2024_31_jan_morning Errors In Measurement
If the percentage errors in measuring the length and the diameter of a wire are 0.1\% each. The percentage error in measuring its resistance will be:
  • A. 0.2\%
  • B. 0.3\%
  • C. 0.1\%
  • D. 0.144\%

Solution

### Related Formula R = fracrho LA = fracrho Lpi left(fracd2right)^2 = frac4rho Lpi d^2 ### Core Logic To find the maximum percentage error in resistance, apply logarithmic differentiation: fracDelta RR = fracDelta LL + 2fracDelta dd Given percentage errors: fracDelta LL times 100\% = 0.1\% fracDelta dd times 100\% = 0.1\% ### Step 2: Substitution Substituting the values: fracDelta RR times 100\% = 0.1\% + 2(0.1\%)\, = 0.1\% + 0.2\% = 0.3\% ### Pattern Recognition Resistance scales inversely with the square of the diameter. The error multiplier for diameter is 2. Just sum linear components directly: Error = L_error + 2 * d_error. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units And Measurements Class 12 Physics: Current Electricity
Q41 jee_main_2024_31_jan_morning Dimensional Analysis
A force is represented by F = ax^2 + bt^1/2 Where x = distance and t = time. The dimensions of b^2 / a are:
  • A. [ML^3T^-3]
  • B. [MLT^-2]
  • C. [ML^-1T^-1]
  • D. [ML^2T^-3]

Solution

### Related Formula textPrinciple of Homogeneity: [F] = [ax^2] = [bt^1/2] ### Core Logic By the principle of dimensional homogeneity, each additive term must have the same dimension as the left hand side. Dimension of force F = [M L T^-2]. For the term ax^2: [a] = frac[F][x^2] = frac[M L T^-2][L^2] = [M L^-1 T^-2] For the term bt^1/2: [b] = frac[F][t^1/2] = frac[M L T^-2][T^1/2] = [M L T^-5/2] ### Step 2: Computing Required Ratio We need the dimension of fracb^2a: left[ fracb^2a right] = frac[M L T^-5/2]^2[M L^-1 T^-2] left[ fracb^2a right] = frac[M^2 L^2 T^-5][M L^-1 T^-2] left[ fracb^2a right] = [M^2-1 L^2 - (-1) T^-5 - (-2)] left[ fracb^2a right] = [M L^3 T^-3] ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Physics: Units And Measurements

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