Three particles each of mass `m` are kept at vertices of an equilateral triangle of side L. The gravitational field at centre due to these particle is

Options

A. zero
B. `(3GM)/(L^(2))`
C. `(9 GM)/(L^(2))`
D. `(12 GM)/(sqrt(3)L^(2))`

Explanation

To find the gravitational field at the center of an equilateral triangle formed by three particles of mass \( m \) at its vertices, we can follow these steps:

### Step 1: Understand the Setup
We have three particles, each of mass \( m \), positioned at the vertices of an equilateral triangle with side length \( L \). The center of the triangle is the centroid, which is the point where the medians intersect.

### Step 2: Calculate the Distance from the Center to Each Vertex
For an equilateral triangle, the distance \( r \) from the centroid to each vertex can be calculated using the formula:
\[ r = \frac{L}{\sqrt{3}} \]

### Step 3: Calculate the Gravitational Field Due to One Particle
The gravitational field \( E \) due to a single particle of mass \( m \) at a distance \( r \) is given by:
\[ E = \frac{Gm}{r^2} \]
Substituting \( r = \frac{L}{\sqrt{3}} \):
\[ E = \frac{Gm}{\left(\frac{L}{\sqrt{3}}\right)^2} = \frac{Gm \cdot 3}{L^2} = \frac{3Gm}{L^2} \]

### Step 4: Determine the Direction of the Gravitational Fields
Each of the three particles exerts a gravitational field at the center directed towards itself. Since the triangle is symmetric, the directions of the gravitational fields from each particle will be at angles of \( 120^\circ \) to each other.

### Step 5: Calculate the Net Gravitational Field
Let \( E_A, E_B, E_C \) be the gravitational fields due to particles at vertices A, B, and C respectively. Since these fields are equal in magnitude and symmetrically arranged, we can use vector addition.

The angle between any two gravitational fields is \( 120^\circ \). The resultant of three vectors of equal magnitude \( E \) at \( 120^\circ \) to each other can be shown to cancel out. Thus:
\[ E_{\text{net}} = E_A + E_B + E_C = 0 \]

### Conclusion
The net gravitational field at the center of the triangle due to the three particles is zero.