The total flux ( in S.I units ) through a closed surface constructed around a positive charge of 0.5 C placed in a dielectric medium of dielectric constant 10 is

Options

A. `5.65xx10^9`
B. `1.13xx10^(11)`
C. `9xx10^9`
D. `8.85xx10^(-12)`

Answer

To solve the problem of finding the total electric flux through a closed surface surrounding a positive charge of 0.5 C placed in a dielectric medium with a dielectric constant of 10, we can follow these steps:

### Step 1: Understand Gauss's Law
Gauss's Law states that the total electric flux (Φ) through a closed surface is proportional to the charge (Q) enclosed by that surface. The formula is given by:

\[ \Phi = \frac{Q}{\epsilon} \]

Where:
- \( \Phi \) is the electric flux,
- \( Q \) is the charge enclosed,
- \( \epsilon \) is the permittivity of the medium.

### Step 2: Identify the Permittivity
In a dielectric medium, the permittivity \( \epsilon \) is given by:

\[ \epsilon = k \cdot \epsilon_0 \]

Where:
- \( k \) is the dielectric constant of the medium (in this case, \( k = 10 \)),
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{F/m} \).

### Step 3: Calculate the Permittivity
Substituting the values into the equation for permittivity:

\[ \epsilon = 10 \cdot (8.85 \times 10^{-12}) = 8.85 \times 10^{-11} \, \text{F/m} \]

### Step 4: Calculate the Electric Flux
Now, we can substitute the values into Gauss's Law to find the total flux:

\[ \Phi = \frac{Q}{\epsilon} = \frac{0.5}{8.85 \times 10^{-11}} \]

### Step 5: Perform the Calculation
Calculating the above expression:

\[ \Phi = \frac{0.5}{8.85 \times 10^{-11}} \approx 5.649 \times 10^{10} \, \text{N m}^2/\text{C} \]

### Step 6: Express in Standard Form
To express the answer in a more manageable form, we can write:

\[ \Phi \approx 5.649 \times 10^{10} \, \text{N m}^2/\text{C} = 0.05649 \times 10^{12} \, \text{N m}^2/\text{C} \]

### Step 7: Final Answer
The final answer can be expressed as:

\[ \Phi \approx 5.65 \times 10^{9} \, \text{N m}^2/\text{C} \]

Thus, the total electric flux through the closed surface is approximately \( 5.65 \times 10^{9} \, \text{N m}^2/\text{C} \).