Infinite bodies, each of mass 3kg are situated at distance 1m,2m,4m,8m...respectively on x-axis. The resultant intensity of gravitational field at the origin will be

Options

A. `G`
B. `2G`
C. `3G`
D. `4G`

Explanation

To find the resultant intensity of the gravitational field at the origin due to an infinite number of bodies, each with a mass of 3 kg located at distances of 1m, 2m, 4m, 8m, and so on along the x-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Gravitational Field**:
The gravitational field \( g \) due to a mass \( m \) at a distance \( r \) is given by the formula:
\[ g = \frac{Gm}{r^2} \]
where \( G \) is the gravitational constant.

2. **Identifying the Masses and Distances**:
The masses are located at distances \( r = 1m, 2m, 4m, 8m, \ldots \). Each mass \( m = 3 \, \text{kg} \).

3. **Calculating the Gravitational Field from Each Mass**:
The gravitational field at the origin due to each mass can be calculated as follows:
- For the mass at \( r = 1m \):
\[ g_1 = \frac{G \cdot 3}{1^2} = 3G \]
- For the mass at \( r = 2m \):
\[ g_2 = \frac{G \cdot 3}{2^2} = \frac{3G}{4} \]
- For the mass at \( r = 4m \):
\[ g_3 = \frac{G \cdot 3}{4^2} = \frac{3G}{16} \]
- For the mass at \( r = 8m \):
\[ g_4 = \frac{G \cdot 3}{8^2} = \frac{3G}{64} \]

4. **Summing the Gravitational Fields**:
The total gravitational field \( g_{\text{total}} \) at the origin is the sum of the gravitational fields due to all the masses:
\[ g_{\text{total}} = g_1 + g_2 + g_3 + g_4 + \ldots \]
This can be expressed as:
\[ g_{\text{total}} = 3G \left(1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots\right) \]

5. **Identifying the Series**:
The series inside the parentheses is a geometric series where the first term \( a = 1 \) and the common ratio \( r = \frac{1}{4} \):
\[ S = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots \]
The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

6. **Calculating the Total Gravitational Field**:
Substituting the sum back into the equation for \( g_{\text{total}} \):
\[ g_{\text{total}} = 3G \cdot \frac{4}{3} = 4G \]

### Final Result:
The resultant intensity of the gravitational field at the origin is:
\[ g_{\text{total}} = 4G \]