Dimensional Analysis

In given equation, should be dimensionless


And

Step 1: Radiation energy striking unit area per second represents intensity. [Q] = EnergyArea × Time = ML²T⁻²L²T = MT⁻³
Step 2: Speed of light has dimensions [c] = LT⁻¹
Step 3: Radiation pressure is given by P = (Q)/(c) [P] = MT⁻³LT⁻¹ = ML⁻¹T⁻²
Step 4: For PˣQʸcᶻ to be dimensionless: (ML⁻¹T⁻²)ˣ(MT⁻³)ʸ(LT⁻¹)ᶻ = M⁰L⁰T⁰ Equating powers: aligned M &: x + y = 0
L &: -x + z = 0
T &: -2x -3y - z = 0 aligned
Step 5: Solving: y = -x, z = x Taking x = 1: y = -1, z = 1

Substitute dimensions of all quantities and equate powers of L to find dimension of q.

Step 1: Formula for Energy.
Energy is given by: where is the force and is the distance.

Step 2: Increase in units.
If both the force and length are increased by 4 times, the energy will increase by times. Thus, the correct answer is (1).

Step 1: Use dimensional analysis.
The dimensions of velocity are . The dimensions of tension are , area is , and density is .
Step 2: Set up the relation.
Let . By equating the dimensions, we solve for , , and to get the expression: Final Answer:

Step 1: Dimensional formula:
Step 2:
Step 3: Equating dimensions: Solving:

Step 1: Dimensional formula Capacitance: [C] = [M⁻1L⁻2T⁴A²] Magnetic induction: [B] = [MT⁻2A⁻1]
Step 2: From X = 3YZ² [Y] = ([X])/([Z]²)
Step 3: Substituting dimensions gives option (A).

Step 1: Dimensions: [C] = LT⁻1, [g] = LT⁻2
Step 2: [(C²)/(g)] = dfracL²T⁻2LT⁻2 = L

Step 1: Dimensions: [p]=1, [l]=L, [F]=MLT⁻2, [m]=M
Step 2: [√((F)/(m))] =√fracMLT⁻2M = L¹/2T⁻1
Step 3: [ν]=(1)/(L)× L¹/2T⁻1 = L⁻1/2T⁻1 Hence the closest dimensional form is: [M⁰L⁻1T⁻1]

Step 1: Given relation: X = 3YZ² ⟹ [Y] = ([X])/([Z]²) Step 2: Dimensions: [X] = capacitance = [M⁻1L⁻2T⁴A²] [Z] = magnetic induction = [MT⁻2A⁻1] Step 3: Substitute: [Y] = fracM⁻1L⁻2T⁴A²M²T⁻4A⁻2 [Y] = M⁻3L⁻2T⁸A⁴

Step 1: Understanding the Concept:
To find the dimensions of the final expression, we must substitute the dimensional formulas for each constituent physical quantity.

Step 2: Key Formula or Approach:
The dimensions are: - (Energy) = - (Angular Momentum) = - (Mass) = - (Gravitational Constant) =

Step 3: Detailed Explanation:
Substitute the dimensions into : Expand the terms: Combine the numerator and denominator: Divide:

Step 4: Final Answer
The expression is dimensionless, so the dimensions are .