What are the units of `K (1)/(4pi in_(0))` ?

Options

A. `C^(2) N^(-1) m^(-2)`
B. `C^(-2) N^(-1) m^(2)`
C. `C^(2) N^(1) m^(2)`
D. unitless

Answer

To find the units of \( K = \frac{1}{4\pi \epsilon_0} \), we can follow these steps:

### Step 1: Understand the context
The constant \( \epsilon_0 \) is known as the permittivity of free space, and it plays a crucial role in electromagnetism, particularly in Coulomb's law, which describes the force between two point charges.

### Step 2: Recall the formula for electric force
Coulomb's law states that the electric force \( F \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
From this equation, we can see that the term \( \frac{1}{4\pi \epsilon_0} \) has units that must balance the units of force and the charges involved.

### Step 3: Identify the units of force
The unit of force \( F \) is the Newton (N). In terms of base units, we have:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2 = \text{M} \cdot \text{L} \cdot \text{T}^{-2} \]

### Step 4: Identify the units of charge
The unit of charge in the SI system is the Coulomb (C).

### Step 5: Write the units of \( \epsilon_0 \)
From the equation of Coulomb's law, we can rearrange it to express \( \epsilon_0 \):
\[ \epsilon_0 = \frac{q_1 q_2}{F \cdot r^2} \]
Thus, the units of \( \epsilon_0 \) can be derived as follows:
- The units of \( q_1 \) and \( q_2 \) are Coulombs (C).
- The units of \( F \) are Newtons (N).
- The units of \( r^2 \) are \( \text{m}^2 \) or \( \text{L}^2 \).

So, the units of \( \epsilon_0 \) can be expressed as:
\[ \text{Units of } \epsilon_0 = \frac{\text{C}^2}{\text{N} \cdot \text{m}^2} = \frac{\text{C}^2}{\text{M} \cdot \text{L} \cdot \text{T}^{-2} \cdot \text{L}^2} = \frac{\text{C}^2}{\text{M} \cdot \text{L}^3 \cdot \text{T}^{-2}} \]

### Step 6: Find the units of \( \frac{1}{\epsilon_0} \)
Taking the reciprocal gives us:
\[ \text{Units of } \frac{1}{\epsilon_0} = \frac{\text{M} \cdot \text{L}^3 \cdot \text{T}^{-2}}{\text{C}^2} \]

### Step 7: Determine the units of \( K = \frac{1}{4\pi \epsilon_0} \)
Since \( 4\pi \) is a dimensionless constant, the units of \( K \) are the same as the units of \( \frac{1}{\epsilon_0} \):
\[ \text{Units of } K = \frac{\text{M} \cdot \text{L}^3 \cdot \text{T}^{-2}}{\text{C}^2} \]

### Final Answer
Thus, the units of \( K = \frac{1}{4\pi \epsilon_0} \) are:
\[ \text{M} \cdot \text{L}^3 \cdot \text{T}^{-2} \cdot \text{C}^{-2} \]

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