Question 14

√(μ0/ε0) has dimensions of impedance (ohm): [M L² T⁻³ A⁻²]. 1 SI unit = 1 kg·m²·s⁻³·A⁻². In new units: mass unit 5 kg, length 5 m, time 5 s, current 5 A. n₂ = n₁ · (1/5)¹ · (1/5)² · (1/5)⁻³ · (1/5)⁻² = 5^(-1-2+3+2) = 5² = 25.

No written reasoning was returned for this item.

**Calculating physics conversion** I'm tackling a physics problem where I need to analyze a new system of units: 5 kg, 5 m, 5 s, and 5 A. The question asks for the magnitude of one SI unit of sqrt(μ0/ε0), which relates to the impedance of free space (ohms). So, I need to express one SI unit in these new units and figure out how that conversion works. I have to ensure I'm careful with the wording and calculations while figuring out the magnitude. **Finding derived unit magnitude** I'm working on dimensions where M^a L^b T^c I^d equals M^1 L^2 T^-3 I^-2, and I need to derive a new unit based on 5 kg, 5 m, 5 s, and 5 A. The calculations show that the new derived unit leads to 5 raised to different powers, ultimately simplifying to 1/25 SI ohm. This means one SI unit equals 25 new units in the new system. The requested magnitude is likely 25, but I want to confirm that I'm not accidentally including extra numeric values.

I need the dimension of √(μ0/ε0), which represents resistance, with units ML²T⁻³A⁻². This confirms that quantity equals the impedance of free space, roughly 377 ohms. Since the new system scales all base units by 5, converting from SI I get n2 = (1/5)^(1+2-3-2) = (1/5)^(-2) = 25. **Answer: 25** The quantity √(μ₀/ε₀) has SI units of resistance (Ohm), with dimensional formula [M¹L²T⁻³A⁻²]. Using dimensional analysis to convert 1 SI unit (in kg, m, s, A) to the new unit system (5 kg, 5 m, 5 s, 5 A): n₂ = n₁ × (M₁/M₂)^1 × (L₁/L₂)^2 × (T₁/T₂)^(-3) × (A₁/A₂)^(-2) n₂ = 1 × (1/5)^1 × (1/5)^2 × (1/5)^(-3) × (1/5)^(-2) n₂ = (1/5)^(1+2-3-2) = (1/5)^(-2) = 25 ```json