Show that the unit of `1/sqrt(epsilon o mu o)` is `(m//s)`.

Explanation

To show that the unit of \( \frac{1}{\sqrt{\epsilon_0 \mu_0}} \) is \( \text{m/s} \), we can follow these steps:

### Step 1: Understand the Constants
- \( \epsilon_0 \) is the permittivity of free space, and its unit is farads per meter (F/m).
- \( \mu_0 \) is the permeability of free space, and its unit is henries per meter (H/m).

### Step 2: Write the Units of \( \epsilon_0 \) and \( \mu_0 \)
- The unit of \( \epsilon_0 \) can be expressed as:
\[ [\epsilon_0] = \text{F/m} = \frac{\text{C}^2}{\text{N} \cdot \text{m}^2} = \frac{\text{C}^2 \cdot \text{s}^2}{\text{kg} \cdot \text{m}^3} \]
(using \( \text{N} = \text{kg} \cdot \text{m/s}^2 \))

- The unit of \( \mu_0 \) can be expressed as:
\[ [\mu_0] = \text{H/m} = \frac{\text{kg}}{\text{C}^2 \cdot \text{s}^2} \cdot \text{m} = \frac{\text{kg}}{\text{C}^2 \cdot \text{s}^2 \cdot \text{m}} \]

### Step 3: Combine the Units
Now, we need to find the unit of \( \epsilon_0 \mu_0 \):
\[ [\epsilon_0 \mu_0] = \left(\frac{\text{C}^2 \cdot \text{s}^2}{\text{kg} \cdot \text{m}^3}\right) \cdot \left(\frac{\text{kg}}{\text{C}^2 \cdot \text{s}^2 \cdot \text{m}}\right) \]

### Step 4: Simplify the Expression
When we multiply these two units, we get:
\[ [\epsilon_0 \mu_0] = \frac{\text{C}^2 \cdot \text{s}^2 \cdot \text{kg}}{\text{kg} \cdot \text{m}^3 \cdot \text{C}^2 \cdot \text{s}^2 \cdot \text{m}} = \frac{1}{\text{m}^2} \]

### Step 5: Take the Square Root
Now, we take the square root of \( \epsilon_0 \mu_0 \):
\[ \sqrt{\epsilon_0 \mu_0} = \sqrt{\frac{1}{\text{m}^2}} = \frac{1}{\text{m}} \]

### Step 6: Find the Unit of \( \frac{1}{\sqrt{\epsilon_0 \mu_0}} \)
Now, we can find the unit of \( \frac{1}{\sqrt{\epsilon_0 \mu_0}} \):
\[ \frac{1}{\sqrt{\epsilon_0 \mu_0}} = \frac{1}{\frac{1}{\text{m}}} = \text{m} \]

### Step 7: Relate to Speed of Light
In electromagnetism, it is known that:
\[ c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \]
where \( c \) is the speed of light. The unit of speed is:
\[ \text{m/s} \]

### Conclusion
Thus, we have shown that the unit of \( \frac{1}{\sqrt{\epsilon_0 \mu_0}} \) is indeed \( \text{m/s} \).

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