The dimensional formula for permittivity of free space ε₀ in the equation F=1/(4π ε₀)·(q₁q₂/r²) where, symbols have their usual meanings is

• A. [M¹L³A⁻²T⁻⁴]
• B. [M⁻¹L⁻³T⁴A²]
• C. [M⁻¹L⁻³A⁻²T⁻⁴]
• D. [M¹L³A²T⁻⁴]

The dimensional formula for the permittivity of free space (ε₀) can be derived from the equation for the electrostatic force, given by Coulomb's law: F = 1/(4π ε₀) · (q₁q₂/r²). In this equation, F represents force, q₁ and q₂ are charges, and r is the distance between the charges.

Breaking Down the Units

To find the dimensional formula for ε₀, we start with the dimensions of the other quantities involved:

• Force (F): [M L T⁻²]
• Charge (q): [A T]
• Distance (r): [L]

Setting Up the Equation

From the equation, we can rearrange it to isolate ε₀:

ε₀ = 1/(4π) · (q₁q₂)/(F · r²)

Substituting Dimensions

Now, substituting the dimensions into the equation:

ε₀ = (A² T²) / ([M L T⁻²] · [L²])

Calculating the Result

This simplifies to:

ε₀ = (A² T²) / ([M L³ T⁻²]) = [M⁻¹ L⁻³ T⁴ A²]

Final Answer

Thus, the correct dimensional formula for the permittivity of free space ε₀ is:

B. [M⁻¹ L⁻³ T⁴ A²]