Two point charges one of `+100 muC` and another of `-400 muC`, are kept 30 cm apart. Find the point of zero potential on the line joining the two charges.

Explanation

To find the point of zero potential between two point charges, we can follow these steps:

### Given:
- Charge \( Q_1 = +100 \, \mu C = +100 \times 10^{-6} \, C \)
- Charge \( Q_2 = -400 \, \mu C = -400 \times 10^{-6} \, C \)
- Distance between the charges \( d = 30 \, cm = 0.3 \, m \)

### Step 1: Set up the problem
Let the distance from the charge \( Q_1 \) (the positive charge) to the point where the potential is zero be \( x \). Therefore, the distance from the charge \( Q_2 \) (the negative charge) to this point will be \( d - x = 0.3 - x \).

### Step 2: Write the expression for electric potential
The electric potential \( V \) at a point due to a point charge is given by:
\[ V = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r} \]
where \( Q \) is the charge and \( r \) is the distance from the charge to the point.

The total potential \( V \) at the point where we want to find zero potential is the sum of the potentials due to both charges:
\[ V = V_1 + V_2 = \frac{1}{4 \pi \epsilon_0} \left( \frac{Q_1}{x} + \frac{Q_2}{0.3 - x} \right) \]

### Step 3: Set the total potential to zero
Setting the total potential \( V \) to zero gives us:
\[ \frac{Q_1}{x} + \frac{Q_2}{0.3 - x} = 0 \]
Substituting the values of \( Q_1 \) and \( Q_2 \):
\[ \frac{100 \times 10^{-6}}{x} + \frac{-400 \times 10^{-6}}{0.3 - x} = 0 \]

### Step 4: Solve the equation
Rearranging the equation:
\[ \frac{100}{x} = \frac{400}{0.3 - x} \]
Cross-multiplying gives:
\[ 100(0.3 - x) = 400x \]
Expanding this:
\[ 30 - 100x = 400x \]
Combining like terms:
\[ 30 = 500x \]
Thus,
\[ x = \frac{30}{500} = 0.06 \, m \]

### Step 5: Convert to centimeters
To convert meters to centimeters:
\[ x = 0.06 \, m = 6 \, cm \]

### Step 6: Find the distance from the second charge
The distance from the second charge \( Q_2 \) is:
\[ 0.3 - x = 0.3 - 0.06 = 0.24 \, m = 24 \, cm \]

### Conclusion
The point of zero potential is located \( 6 \, cm \) from the charge \( +100 \, \mu C \) and \( 24 \, cm \) from the charge \( -400 \, \mu C \).

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