(A ): A charge is lying at the centre of the line joining two similar charges each which are fixed. The system will be in equilibrium if that charge is one fourth of the similar charges.
(R ): For charge to be in equilibrium, sum of the forces on charge due to rest of the two charges must be zero.

Options

A. Both .A. and .R. are true and .R. is the correct explanation of .A.
B. Both .A. and .R. are tru and .R. is not the correct explanation of .A.
C. A. is true and .R. is false
D. A. is false and .R. is true

Answer

To solve the problem, we need to analyze the situation described in the question step by step.

### Step 1: Understand the Configuration
We have two fixed charges, let's denote them as \( Q \) and \( Q \), which are identical and fixed at two points. There is a third charge, which we will denote as \( q \), placed at the midpoint between the two charges \( Q \).

### Step 2: Determine the Forces Acting on Charge \( q \)
The charge \( q \) experiences forces due to both charges \( Q \). Since both charges are identical, the forces acting on \( q \) from each charge will be equal in magnitude but opposite in direction if \( q \) is placed exactly in the middle.

### Step 3: Calculate the Forces
The force \( F \) on charge \( q \) due to one of the charges \( Q \) can be calculated using Coulomb's Law:

\[ F = k \frac{|Q \cdot q|}{r^2} \]

where \( k \) is Coulomb's constant, and \( r \) is the distance from charge \( Q \) to charge \( q \). Since \( q \) is at the midpoint, the distance \( r \) is half the distance between the two charges.

### Step 4: Set Up the Condition for Equilibrium
For the charge \( q \) to be in equilibrium, the net force acting on it must be zero. This means that the force exerted by the left charge \( Q \) on \( q \) must equal the force exerted by the right charge \( Q \) on \( q \):

\[ F_{left} = F_{right} \]

Since both forces are equal in magnitude, we can write:

\[ k \frac{|Q \cdot q|}{r^2} = k \frac{|Q \cdot q|}{r^2} \]

This condition is satisfied as long as the magnitudes of the forces are equal.

### Step 5: Analyze the Condition for Specific Charge \( q \)
Now, we need to find the specific value of \( q \) that will satisfy this equilibrium condition. According to the assertion, it is claimed that \( q \) should be one fourth of \( Q \):

\[ q = \frac{1}{4} Q \]

### Step 6: Substitute and Verify
Substituting \( q = \frac{1}{4} Q \) into the force equation, we can check if the forces balance:

\[ F = k \frac{|Q \cdot \frac{1}{4} Q|}{r^2} = k \frac{\frac{1}{4} Q^2}{r^2} \]

This force will be equal from both sides since both charges \( Q \) are identical.

### Conclusion
Thus, the assertion that the system will be in equilibrium if the charge \( q \) is one fourth of the similar charges \( Q \) is indeed **false** because the forces will not balance unless \( q \) is zero or equal to \( Q \). The reason that the sum of the forces must be zero is **true**.

### Final Answer
- Assertion (A): False
- Reason (R): True