Two unlike charges of magnitude `q` are separated by a distance `2d`. The potential at a point midway between them is

Options

A. Zero
B. `(1)/(4pi epsilon_(0))`
C. `(1)/(4pi epsilon_(0)).(q)/(d)`
D. `(1)/(4pi epsilon_(0)).(2q)/(d^(2))`

Explanation

To find the electric potential at a point midway between two unlike charges of magnitude \( q \) separated by a distance \( 2d \), we can follow these steps:

### Step 1: Identify the Charges and Their Positions
Let’s denote the two charges as \( +q \) and \( -q \). Place the positive charge \( +q \) at position \( A \) and the negative charge \( -q \) at position \( B \). The distance between the two charges is \( 2d \).

- Position of \( +q \) (Charge 1): \( A \) (let's say at \( x = -d \))
- Position of \( -q \) (Charge 2): \( B \) (let's say at \( x = +d \))

### Step 2: Determine the Midpoint
The midpoint \( P \) between the two charges is located at \( x = 0 \). The distance from the midpoint to each charge is \( d \).

### Step 3: Calculate the Potential Due to Each Charge
The electric potential \( V \) due to a point charge is given by the formula:
\[ V = \frac{k \cdot q}{r} \]
where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point where the potential is being calculated.

- **Potential at point \( P \) due to charge \( +q \)**:
\[ V_1 = \frac{k \cdot (+q)}{d} \]

- **Potential at point \( P \) due to charge \( -q \)**:
\[ V_2 = \frac{k \cdot (-q)}{d} \]

### Step 4: Calculate the Total Potential at the Midpoint
The total electric potential \( V \) at point \( P \) is the algebraic sum of the potentials due to both charges:
\[ V = V_1 + V_2 \]
Substituting the values we calculated:
\[ V = \frac{k \cdot (+q)}{d} + \frac{k \cdot (-q)}{d} \]
\[ V = \frac{kq}{d} - \frac{kq}{d} \]
\[ V = 0 \]

### Conclusion
The electric potential at the midpoint between the two unlike charges is \( 0 \).

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