One Planck length is equal to

Options

A. `10^(-12)` m
B. `10^(-25)` m
C. `10^(-35)` m
D. `10^(-40)` m

Answer

To find the value of one Planck length, we can follow these steps:

### Step 1: Understand the Definition
Planck length is considered the smallest measurable length in the universe and is derived from fundamental physical constants.

### Step 2: Use the Formula
The formula for Planck length (Lp) is given by:
\[ L_p = \sqrt{\frac{h}{2 \pi g c^3}} \]
where:
- \( h \) = Planck's constant
- \( g \) = gravitational constant
- \( c \) = speed of light

### Step 3: Substitute the Constants
We need to substitute the known values of the constants:
- \( h \approx 6.626 \times 10^{-34} \, \text{Js} \)
- \( g \approx 6.674 \times 10^{-11} \, \text{m}^3/\text{kg s}^2 \)
- \( c \approx 3.00 \times 10^8 \, \text{m/s} \)

### Step 4: Calculate the Planck Length
Now, substituting the values into the formula:
\[ L_p = \sqrt{\frac{6.626 \times 10^{-34}}{2 \pi (6.674 \times 10^{-11}) (3.00 \times 10^8)^3}} \]

### Step 5: Simplify the Calculation
First, calculate \( c^3 \):
\[ c^3 = (3.00 \times 10^8)^3 = 2.7 \times 10^{25} \, \text{m}^3/\text{s}^3 \]
Now substitute this back into the equation:
\[ L_p = \sqrt{\frac{6.626 \times 10^{-34}}{2 \pi (6.674 \times 10^{-11}) (2.7 \times 10^{25})}} \]

### Step 6: Final Calculation
After performing the calculations, we find:
\[ L_p \approx 1.6 \times 10^{-35} \, \text{meters} \]

### Conclusion
Thus, one Planck length is approximately:
\[ L_p \approx 1.6 \times 10^{-35} \, \text{meters} \]