An infinitely long thin straight wire has uniform linear charge density of 1/4 coul `m^(-1)`. Then the magnitude of the electric intensity at a point 18 cm away is

Options

A. `0.33 xx 10^(11)NC^(-1)`
B. `3 xx 10^(11) NC^(-1)`
C. `0.25xx 10^(11)NC^(-1)`
D. `1.32 xx 10^(11)NC^(-1)`

Answer

To solve the problem of finding the electric field intensity at a point 18 cm away from an infinitely long thin straight wire with a uniform linear charge density of \( \lambda = \frac{1}{4} \, \text{C/m} \), we can follow these steps:

### Step 1: Understand the Formula for Electric Field Intensity
The electric field intensity \( E \) due to an infinitely long straight wire with linear charge density \( \lambda \) is given by the formula:
\[ E = \frac{\lambda}{2 \pi \epsilon_0 r} \]
where:
- \( E \) is the electric field intensity,
- \( \lambda \) is the linear charge density,
- \( \epsilon_0 \) is the permittivity of free space, approximately \( 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \),
- \( r \) is the perpendicular distance from the wire.

### Step 2: Convert Units
Convert the distance from centimeters to meters:
\[ r = 18 \, \text{cm} = 0.18 \, \text{m} \]

### Step 3: Substitute Values into the Formula
Substituting the values into the formula:
- \( \lambda = \frac{1}{4} \, \text{C/m} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)
- \( r = 0.18 \, \text{m} \)

The electric field intensity becomes:
\[ E = \frac{\frac{1}{4}}{2 \pi (8.85 \times 10^{-12}) (0.18)} \]

### Step 4: Calculate the Electric Field Intensity
Calculating the denominator:
\[ 2 \pi \epsilon_0 r = 2 \pi (8.85 \times 10^{-12}) (0.18) \]
Calculating this gives:
\[ 2 \pi (8.85 \times 10^{-12}) (0.18) \approx 1.005 \times 10^{-10} \, \text{N m}^2/\text{C}^2 \]

Now substituting back into the equation for \( E \):
\[ E = \frac{0.25}{1.005 \times 10^{-10}} \approx 2.487 \times 10^{9} \, \text{N/C} \]

### Step 5: Final Result
Thus, the magnitude of the electric field intensity at a point 18 cm away from the wire is approximately:
\[ E \approx 2.487 \times 10^{9} \, \text{N/C} \]