Speed of light in vacuum is given by $3\times {10}^8{\mathrm{ms}}^{-1}$ in MKS system. If in a new system, the unit of length and time is increased by a factor of 2 and 4 respectively then find the numerical value of speed of light in this new system (in term of $\times {10}^8$).

The speed of light in vacuum in the MKS system is given as 3 × 10⁸ m/s. Now, let us consider a new system where the units of length and time are scaled by factors of 2 and 4, respectively. Our goal is to find the numerical value of the speed of light in vacuum in this new system, expressed in terms of 10⁸ m/s.

To solve this, we use the following relationship between the quantities in the old and new systems:

    n₁ u₁ = n₂ u₂

Where:

• n₁ is the speed of light in vacuum in the original MKS system (3 × 10⁸ m/s).
• u₁ is the unit of length in the original MKS system.
• n₂ is the speed of light in vacuum in the new system (which we need to find).
• u₂ is the unit of length in the new system.

The relationship for the speed of light in vacuum becomes:

    n₂ = (u₁ / u₂) × n₁

Now, the units of length and time in the new system are given as:

• The unit of length is increased by a factor of 2.
• The unit of time is increased by a factor of 4.

Substituting these values into the equation:

    n₂ = (L₁ / L₂) × (T₂ / T₁) × n₁

Where:

• L₁ is the unit of length in the original system.
• T₁ is the unit of time in the original system.
• L₂ is the unit of length in the new system (2 times L₁).
• T₂ is the unit of time in the new system (4 times T₁).

Now, substitute the given scaling factors into the formula:

    n₂ = (L₁ / (2L₁)) × (4T₁ / T₁) × (3 × 108)

Simplifying the expression:

    n₂ = (1/2) × 4 × (3 × 108) = 6 × 108 m/s

Thus, the speed of light in vacuum in the new system is 6 × 10⁸ m/s.