Three charged particles having charges q, -2q & q are placed in a line at points (-a, 0), (0,0) & (a , 0) respectively. The expression for electric potential at P(r, 0 ) for r `gt gt ` a is

Options

A. `(1)/(4 pi epsilon_(0))(qa^(2))/(r^(4))`
B. `(1)/(4 pi epsilon_(0))(2qa^(2))/(r^(3))`
C. `(1)/(4 pi epsilon_(0))(4qa^(2))/(r^(2))`
D. `(1)/(4 pi epsilon_(0))(8qa^(2))/(r )`

Answer

To find the expression for the electric potential at point P(r, 0) due to the three charged particles with charges \( q, -2q, \) and \( q \) located at points \( (-a, 0), (0, 0), \) and \( (a, 0) \) respectively, we will follow these steps:

### Step 1: Write the formula for electric potential
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( k = \frac{1}{4\pi \epsilon_0} \).

### Step 2: Calculate the distance from each charge to point P
For point P located at \( (r, 0) \):
- The distance from charge \( q \) at \( (-a, 0) \) to point P is:
\[ d_1 = r + a \]
- The distance from charge \( -2q \) at \( (0, 0) \) to point P is:
\[ d_2 = r \]
- The distance from charge \( q \) at \( (a, 0) \) to point P is:
\[ d_3 = r - a \]

### Step 3: Write the potentials due to each charge
Now we can express the potential at point P due to each charge:
1. Due to charge \( q \) at \( (-a, 0) \):
\[ V_1 = \frac{kq}{r + a} \]
2. Due to charge \( -2q \) at \( (0, 0) \):
\[ V_2 = \frac{-2kq}{r} \]
3. Due to charge \( q \) at \( (a, 0) \):
\[ V_3 = \frac{kq}{r - a} \]

### Step 4: Calculate the total potential at point P
The total electric potential \( V \) at point P is the sum of the potentials due to all three charges:
\[ V = V_1 + V_2 + V_3 \]
Substituting the expressions we found:
\[ V = \frac{kq}{r + a} - \frac{2kq}{r} + \frac{kq}{r - a} \]

### Step 5: Simplify the expression
To simplify this expression, we need to find a common denominator, which is \( (r + a)(r)(r - a) \):
\[ V = kq \left( \frac{r(r - a) + (-2)(r + a)(r - a) + (r + a)r}{(r + a)(r)(r - a)} \right) \]
After simplification and using the approximation \( r \gg a \), we can neglect terms involving \( a \) in the denominator.

### Step 6: Final expression
After performing the algebra and simplifying, we arrive at:
\[ V \approx \frac{2kqa^2}{r^3} \]
Thus, the expression for the electric potential at point P when \( r \gg a \) is:
\[ V \approx \frac{2kqa^2}{r^3} \]