Two point charges `4 mu C and -2 mu C` are separated by a distance of 1m in air. Calculate at what point on the line joining the two charges is the electric potential zero ?

Explanation

To solve the problem of finding the point along the line joining two point charges where the electric potential is zero, we can follow these steps:

### Step 1: Understand the Configuration
We have two point charges:
- Charge \( Q_1 = +4 \, \mu C = 4 \times 10^{-6} \, C \)
- Charge \( Q_2 = -2 \, \mu C = -2 \times 10^{-6} \, C \)

These charges are separated by a distance of \( d = 1 \, m \).

### Step 2: Define the Point of Interest
Let’s denote the point where the electric potential is zero as point \( P \). We will place point \( P \) at a distance \( x \) from the charge \( Q_1 \) (the positive charge). Consequently, the distance from charge \( Q_2 \) (the negative charge) to point \( P \) will be \( (1 - x) \).

### Step 3: Write the Expression for Electric Potential
The electric potential \( V \) at point \( P \) due to both charges is given by the formula:
\[ V = V_1 + V_2 \]
Where:
- \( V_1 \) is the potential due to \( Q_1 \)
- \( V_2 \) is the potential due to \( Q_2 \)

The electric potential due to a point charge is given by:
\[ V = \frac{1}{4\pi \epsilon_0} \cdot \frac{Q}{r} \]
Thus, we have:
\[ V_1 = \frac{1}{4\pi \epsilon_0} \cdot \frac{4 \times 10^{-6}}{x} \]
\[ V_2 = \frac{1}{4\pi \epsilon_0} \cdot \frac{-2 \times 10^{-6}}{1 - x} \]

### Step 4: Set the Total Electric Potential to Zero
To find the point where the total electric potential is zero, we set:
\[ V_1 + V_2 = 0 \]
This gives us:
\[ \frac{4 \times 10^{-6}}{x} - \frac{2 \times 10^{-6}}{1 - x} = 0 \]

### Step 5: Simplify the Equation
Rearranging the equation, we have:
\[ \frac{4}{x} = \frac{2}{1 - x} \]
Cross-multiplying yields:
\[ 4(1 - x) = 2x \]
Expanding this gives:
\[ 4 - 4x = 2x \]
Combining like terms results in:
\[ 4 = 6x \]
Thus, solving for \( x \):
\[ x = \frac{4}{6} = \frac{2}{3} \, m \]

### Step 6: Determine the Position of the Zero Potential Point
The point where the electric potential is zero is located \( \frac{2}{3} \, m \) from the charge \( Q_1 \) and \( \frac{1}{3} \, m \) from the charge \( Q_2 \).

### Final Answer
The electric potential is zero at a point \( \frac{2}{3} \, m \) from the charge \( +4 \, \mu C \) and \( \frac{1}{3} \, m \) from the charge \( -2 \, \mu C \).

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