The speed of light in vacuum is equal to

Options

A. `mu_0epsilon_0`
B. `sqrt(mu_0epsilon_0)`
C. `1/(mu_0epsilon_0)`
D. `1/(sqrt(mu_0epsilon_0))`

Answer

To find the speed of light in a vacuum, we can use the relationship derived from electromagnetic theory. Here’s the step-by-step solution:

### Step 1: Understand the relationship
The speed of light (c) in a vacuum can be expressed in terms of the permeability (μ₀) and permittivity (ε₀) of free space. The formula is given by:

\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

### Step 2: Identify the constants
- **μ₀ (Permeability of free space)**: This is a measure of how much resistance is encountered when forming a magnetic field in a vacuum. Its value is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).
- **ε₀ (Permittivity of free space)**: This is a measure of how much electric field is 'permitted' in a vacuum. Its value is approximately \( 8.854 \times 10^{-12} \, \text{F/m} \).

### Step 3: Substitute the values into the formula
To find the speed of light, we substitute the known values of μ₀ and ε₀ into the equation:

\[ c = \frac{1}{\sqrt{(4\pi \times 10^{-7}) \times (8.854 \times 10^{-12})}} \]

### Step 4: Calculate the product of μ₀ and ε₀
First, calculate the product:

\[ \mu_0 \epsilon_0 = (4\pi \times 10^{-7}) \times (8.854 \times 10^{-12}) \]

### Step 5: Take the square root
Now, take the square root of the product calculated in Step 4.

### Step 6: Calculate the speed of light
Finally, take the reciprocal of the square root to find the speed of light:

\[ c \approx 3 \times 10^8 \, \text{m/s} \]

Thus, the speed of light in a vacuum is approximately \( 3 \times 10^8 \, \text{m/s} \).

### Final Answer
The speed of light in vacuum is equal to \( \frac{1}{\sqrt{\mu_0 \epsilon_0}} \).

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