It is requird to hold four equal point charges `+q` in equilibrium at the corners of a square. Find the point charge that will do this, if placed at the centre of the square.

Options

A. `(1+sqrt(2)/2)q`
B. `((1+2sqrt(2))/4)q`
C. `((1+2sqrt(2))/2)q`
D. `((1+sqrt(2))/(4))q`

Explanation

To solve the problem of holding four equal point charges \( +q \) in equilibrium at the corners of a square by placing a charge at the center, we can follow these steps:

### Step 1: Understand the Forces Acting on the Charges
The four charges \( +q \) at the corners of the square will repel each other due to their like charges. To achieve equilibrium, we need to introduce a charge \( Q \) at the center of the square that will exert an attractive force on the corner charges.

### Step 2: Determine the Geometry
Let the side length of the square be \( a \). The distance from the center of the square to any corner is given by \( r = \frac{a}{\sqrt{2}} \).

### Step 3: Calculate the Force on One Corner Charge
The force \( F \) on one corner charge \( +q \) due to the other three corner charges can be calculated. The force between two point charges is given by Coulomb's law:
\[ F = k \frac{q^2}{r^2} \]
where \( k \) is Coulomb's constant. The distance \( r \) between the corner charges is \( a \).

### Step 4: Calculate the Resultant Force from Two Charges
The net force on one charge due to the other two adjacent charges can be calculated. Since these forces act at right angles to each other, the resultant force \( F_{net} \) can be calculated using the Pythagorean theorem:
\[ F_{net} = \sqrt{F^2 + F^2} = \sqrt{2F^2} = F\sqrt{2} \]
Substituting \( F \):
\[ F_{net} = \sqrt{2} \cdot k \frac{q^2}{a^2} \]

### Step 5: Calculate the Force Exerted by Charge \( Q \)
The force exerted by the charge \( Q \) at the center on one corner charge \( +q \) is given by:
\[ F_Q = k \frac{qQ}{r^2} = k \frac{qQ}{\left(\frac{a}{\sqrt{2}}\right)^2} = k \frac{qQ}{\frac{a^2}{2}} = \frac{2kqQ}{a^2} \]

### Step 6: Set the Forces Equal for Equilibrium
For the charges to be in equilibrium, the attractive force \( F_Q \) must balance the net repulsive force \( F_{net} \):
\[ \frac{2kqQ}{a^2} = \sqrt{2} \cdot k \frac{q^2}{a^2} \]
We can cancel \( k \) and \( a^2 \) from both sides:
\[ 2Q = \sqrt{2} q \]

### Step 7: Solve for Charge \( Q \)
Rearranging the equation gives:
\[ Q = \frac{\sqrt{2}}{2} q = \frac{q}{\sqrt{2}} \]

### Final Answer
The charge \( Q \) that needs to be placed at the center of the square to hold the four equal point charges \( +q \) in equilibrium is:
\[ Q = \frac{q}{\sqrt{2}} \]

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