A rod of length 2.4 m and radius 4.6 mm carries a negative charge of `4.2xx10^(-7)C` spread uniformly over it surface. The electric field near the mid-point of the rod, at a point on its surface is

Options

A. `-8.6xx10^(5)NC^(-1)`
B. `8.6xx10^(4)NC^(-1)`
C. `-6.7xx10^(5)NC^(-1)`
D. `6.7xx10^(4)NC^(-1)`

Explanation

To find the electric field near the midpoint of a uniformly charged rod at a point on its surface, we can follow these steps:

### Step 1: Calculate the Charge per Unit Length (λ)
Given:
- Total charge (Q) = \(4.2 \times 10^{-7} \, C\)
- Length of the rod (L) = \(2.4 \, m\)

The charge per unit length (λ) can be calculated using the formula:
\[ \lambda = \frac{Q}{L} \]

Substituting the values:
\[ \lambda = \frac{4.2 \times 10^{-7}}{2.4} = 1.75 \times 10^{-7} \, C/m \]

### Step 2: Use the Electric Field Formula
The electric field (E) at a point on the surface of a uniformly charged rod can be calculated using the formula:
\[ E = \frac{2K\lambda}{R} \]
where:
- \(K\) = \(9 \times 10^9 \, N \cdot m^2/C^2\) (Coulomb's constant)
- \(R\) = radius of the rod = \(4.6 \, mm = 4.6 \times 10^{-3} \, m\)

### Step 3: Substitute the Values into the Electric Field Formula
Now substituting the values of \(K\), \(\lambda\), and \(R\) into the electric field formula:
\[ E = \frac{2 \times 9 \times 10^9 \times 1.75 \times 10^{-7}}{4.6 \times 10^{-3}} \]

### Step 4: Calculate the Electric Field
Calculating the above expression:
\[ E = \frac{2 \times 9 \times 10^9 \times 1.75 \times 10^{-7}}{4.6 \times 10^{-3}} = \frac{3.15 \times 10^3}{4.6 \times 10^{-3}} \approx 6.83 \times 10^5 \, N/C \]

### Step 5: Consider the Sign of the Electric Field
Since the charge on the rod is negative, the electric field will also be negative, indicating the direction of the field is towards the rod:
\[ E \approx -6.83 \times 10^5 \, N/C \]

### Final Answer
The electric field near the midpoint of the rod, at a point on its surface, is approximately:
\[ E \approx -6.83 \times 10^5 \, N/C \]