The dimensional formula for magnetic permeability μ is

A. [ ML T⁻² A⁻² ]

B. [ M⁰ L⁻¹ T ]

C. [ M⁰ L² T⁻² A⁻² ]

D. [ M L² T⁻² A⁻² ]

To determine the dimensional formula for magnetic permeability (μ), we first need to understand its relationship with other physical quantities. Magnetic permeability is a measure of how much a material responds to a magnetic field, and it is defined in terms of magnetic field strength (H) and magnetic flux density (B). The relationship can be expressed as:

Understanding the Relationship

The relationship between magnetic flux density (B) and magnetic field strength (H) is given by the equation:

B = μH

Here, B is measured in teslas (T), and H is measured in amperes per meter (A/m). To find the dimensional formula for μ, we can rearrange this equation:

μ = B / H

Breaking Down the Dimensions

Next, we need to analyze the dimensions of B and H:

• Magnetic Flux Density (B): The unit of B is the tesla (T), which can be expressed in terms of base SI units as:

1 T = 1 kg/(A·s²)

• Thus, the dimensional formula for B is:

[M L T⁻² A⁻¹]

• Magnetic Field Strength (H): The unit of H is amperes per meter (A/m), which has the dimensional formula:

[M⁰ L⁻¹ T⁰ A¹]

Calculating the Dimensional Formula for μ

Now, substituting the dimensional formulas of B and H into the equation for μ:

μ = B / H

Substituting the dimensions:

μ = [M L T⁻² A⁻¹] / [M⁰ L⁻¹ T⁰ A¹]

When we divide the dimensions, we subtract the exponents of like terms:

• For M: 1 - 0 = 1 → M¹
• For L: 1 - (-1) = 1 + 1 = 2 → L²
• For T: -2 - 0 = -2 → T⁻²
• For A: -1 - 1 = -2 → A⁻²

Thus, the dimensional formula for magnetic permeability μ is:

[M L² T⁻² A⁻²]

Final Answer

From the options provided, the correct choice is:

C. [M⁰ L² T⁻² A⁻²]

It's important to note that the dimensional formula indicates that magnetic permeability is independent of mass (M⁰), while it depends on length squared (L²), time squared (T⁻²), and the inverse square of current (A⁻²).