Infinite number of bodies, each of mass `2kg`, are situated on `x`-axis at distance `1m,2m,4m,8m.........` respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

Options

A. `-G`
B. `-(8)/(3)G`
C. `-(4)/(3)G`
D. `-4G`

Answer

To find the gravitational potential at the origin due to an infinite number of bodies each of mass \(2 \, \text{kg}\) located at distances \(1 \, \text{m}, 2 \, \text{m}, 4 \, \text{m}, 8 \, \text{m}, \ldots\) from the origin, we can follow these steps:

### Step 1: Understand the formula for gravitational potential
The gravitational potential \(V\) due to a point mass \(m\) at a distance \(r\) is given by the formula:
\[ V = -\frac{Gm}{r} \]
where \(G\) is the gravitational constant.

### Step 2: Calculate the potential contributions from each mass
We have masses located at distances that are powers of 2:
- At \(r = 1 \, \text{m}\):
\[ V_1 = -\frac{G \cdot 2}{1} = -2G \]
- At \(r = 2 \, \text{m}\):
\[ V_2 = -\frac{G \cdot 2}{2} = -G \]
- At \(r = 4 \, \text{m}\):
\[ V_3 = -\frac{G \cdot 2}{4} = -\frac{G}{2} \]
- At \(r = 8 \, \text{m}\):
\[ V_4 = -\frac{G \cdot 2}{8} = -\frac{G}{4} \]
- And so on...

### Step 3: Write the total potential at the origin
The total gravitational potential \(V\) at the origin due to all the masses is the sum of the individual potentials:
\[ V = V_1 + V_2 + V_3 + V_4 + \ldots \]
Substituting the values we calculated:
\[ V = -2G - G - \frac{G}{2} - \frac{G}{4} - \ldots \]

### Step 4: Factor out the common term
We can factor out \(-G\) from the sum:
\[ V = -G \left(2 + 1 + \frac{1}{2} + \frac{1}{4} + \ldots\right) \]

### Step 5: Recognize the series as a geometric series
The series inside the parentheses is a geometric series where:
- The first term \(a = 2\)
- The common ratio \(r = \frac{1}{2}\)

The sum \(S\) of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Substituting the values:
\[ S = \frac{2}{1 - \frac{1}{2}} = \frac{2}{\frac{1}{2}} = 4 \]

### Step 6: Substitute back to find the total potential
Now substitute \(S\) back into the potential equation:
\[ V = -G \cdot 4 = -4G \]

### Final Answer
Thus, the resulting gravitational potential at the origin due to this system is:
\[ \boxed{-4G} \]