A charge of `4xx10^(-9)C` is distributed uniformly over the circumference of a conducting ring of radius 0.3m. Calculate the field intensity at a point on the axis of the ring at 0.4m from its centre, and also at the centre.

Explanation

To solve the problem of finding the electric field intensity at a point on the axis of a conducting ring and at its center, we will follow these steps:

### Step 1: Identify the given values
- Charge, \( q = 4 \times 10^{-9} \, \text{C} \)
- Radius of the ring, \( r = 0.3 \, \text{m} \)
- Distance from the center of the ring along the axis, \( x = 0.4 \, \text{m} \)

### Step 2: Use the formula for electric field intensity on the axis of a ring
The electric field intensity \( E \) at a distance \( x \) from the center of a ring with charge \( q \) and radius \( r \) is given by the formula:
\[ E = \frac{k \cdot q \cdot x}{(x^2 + r^2)^{3/2}} \]
where \( k \) is Coulomb's constant, \( k = 9 \times 10^9 \, \text{N m}^2/\text{C}^2 \).

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ E = \frac{(9 \times 10^9) \cdot (4 \times 10^{-9}) \cdot (0.4)}{((0.4)^2 + (0.3)^2)^{3/2}} \]

### Step 4: Calculate \( x^2 + r^2 \)
Calculate \( x^2 + r^2 \):
\[ x^2 = (0.4)^2 = 0.16 \]
\[ r^2 = (0.3)^2 = 0.09 \]
\[ x^2 + r^2 = 0.16 + 0.09 = 0.25 \]

### Step 5: Calculate \( (x^2 + r^2)^{3/2} \)
Now calculate \( (0.25)^{3/2} \):
\[ (0.25)^{3/2} = (0.25)^{1.5} = \sqrt{0.25^3} = \sqrt{0.015625} = 0.125 \]

### Step 6: Substitute back into the electric field formula
Now substitute back into the electric field formula:
\[ E = \frac{(9 \times 10^9) \cdot (4 \times 10^{-9}) \cdot (0.4)}{0.125} \]

### Step 7: Calculate the electric field intensity
Calculating the numerator:
\[ 9 \times 10^9 \cdot 4 \times 10^{-9} \cdot 0.4 = 14.4 \]
Now divide by \( 0.125 \):
\[ E = \frac{14.4}{0.125} = 115.2 \, \text{N/C} \]

### Step 8: Electric field at the center of the ring
At the center of the ring, \( x = 0 \):
\[ E = 0 \, \text{N/C} \]

### Final Results
- Electric field intensity at \( 0.4 \, \text{m} \) from the center of the ring: \( E = 115.2 \, \text{N/C} \)
- Electric field intensity at the center of the ring: \( E = 0 \, \text{N/C} \)