To solve the problem of finding the electric field at the point (a, 0, 0) due to two point charges \( q \) and \( -q \) located at (0, 0, d) and (0, 0, -d) respectively, we can follow these steps:
### Step 1: Understand the Configuration
We have two point charges:
- Charge \( q \) at position \( (0, 0, d) \)
- Charge \( -q \) at position \( (0, 0, -d) \)
We need to find the electric field at the point \( (a, 0, 0) \).
### Step 2: Calculate the Distance from Each Charge to the Point
Using the distance formula, the distance \( r \) from each charge to the point \( (a, 0, 0) \) can be calculated as follows:
1. For charge \( q \):
\[
r_1 = \sqrt{(a - 0)^2 + (0 - 0)^2 + (0 - d)^2} = \sqrt{a^2 + d^2}
\]
2. For charge \( -q \):
\[
r_2 = \sqrt{(a - 0)^2 + (0 - 0)^2 + (0 + d)^2} = \sqrt{a^2 + d^2}
\]
Both distances are equal, \( r_1 = r_2 = \sqrt{a^2 + d^2} \).
### Step 3: Calculate the Electric Field Due to Each Charge
The electric field \( E \) due to a point charge is given by the formula:
\[
E = \frac{k \cdot |q|}{r^2}
\]
where \( k = \frac{1}{4 \pi \epsilon_0} \).
1. Electric field \( E_1 \) due to charge \( q \):
\[
E_1 = \frac{k \cdot q}{r_1^2} = \frac{k \cdot q}{(a^2 + d^2)}
\]
2. Electric field \( E_2 \) due to charge \( -q \):
\[
E_2 = \frac{k \cdot | -q |}{r_2^2} = \frac{k \cdot q}{(a^2 + d^2)}
\]
### Step 4: Determine the Direction of Each Electric Field
- The electric field \( E_1 \) due to charge \( q \) points away from the charge, towards the positive x-direction.
- The electric field \( E_2 \) due to charge \( -q \) points towards the charge, which is also towards the positive x-direction.
### Step 5: Calculate the Net Electric Field
Since both electric fields point in the same direction (positive x-direction), we can add them together:
\[
E_{\text{net}} = E_1 + E_2 = \frac{k \cdot q}{(a^2 + d^2)} + \frac{k \cdot q}{(a^2 + d^2)} = \frac{2k \cdot q}{(a^2 + d^2)}
\]
### Step 6: Final Expression
Thus, the electric field at the point \( (a, 0, 0) \) is:
\[
E_{\text{net}} = \frac{2k \cdot q}{(a^2 + d^2)} \hat{i}
\]
