Two point charges q and `-2q` are kept 'd' distance apart. Find the location of the point relative to charge q at which potential due to the system of charges is zero.

Explanation

To find the location of the point relative to charge \( q \) at which the potential due to the system of charges is zero, we can follow these steps:

### Step 1: Understand the Setup
We have two point charges:
- Charge \( q \) located at point A.
- Charge \( -2q \) located at point B, which is \( d \) distance away from charge \( q \).

### Step 2: Define the Point of Interest
Let’s denote the point where the potential is zero as point P. Let the distance from charge \( q \) to point P be \( x \). Consequently, the distance from charge \( -2q \) to point P will be \( d - x \).

### Step 3: Write the Expression for Electric Potential
The electric potential \( V \) at point P due to both charges can be expressed as:
\[ V_P = V_q + V_{-2q} \]
Where:
- \( V_q = \frac{1}{4\pi \epsilon_0} \cdot \frac{q}{x} \) (Potential due to charge \( q \))
- \( V_{-2q} = \frac{1}{4\pi \epsilon_0} \cdot \frac{-2q}{d - x} \) (Potential due to charge \( -2q \))

### Step 4: Set the Total Potential to Zero
To find the point where the total potential is zero, we set the equation:
\[ \frac{1}{4\pi \epsilon_0} \cdot \frac{q}{x} + \frac{1}{4\pi \epsilon_0} \cdot \frac{-2q}{d - x} = 0 \]
We can simplify this by multiplying through by \( 4\pi \epsilon_0 \):
\[ \frac{q}{x} - \frac{2q}{d - x} = 0 \]

### Step 5: Solve for \( x \)
Now, we can rearrange the equation:
\[ \frac{q}{x} = \frac{2q}{d - x} \]
Cancelling \( q \) (assuming \( q \neq 0 \)):
\[ \frac{1}{x} = \frac{2}{d - x} \]
Cross-multiplying gives:
\[ d - x = 2x \]
Rearranging terms results in:
\[ d = 3x \]
Thus, we find:
\[ x = \frac{d}{3} \]

### Step 6: Determine the Location Relative to Charge \( q \)
The point P where the potential is zero is located at a distance of \( \frac{d}{3} \) from charge \( q \).

### Final Answer
The location of the point relative to charge \( q \) at which the potential due to the system of charges is zero is \( \frac{d}{3} \).

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