Two charges -4 `mu`C and 8 `mu`C are 27 cm apart. The distance from the first charge on line of joining between two charges where electric potential would be zero is

Options

A. 10 cm
B. 9 cm
C. 20 cm
D. 21 cm

Answer

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

### Step 1: Understand the Problem
We have two charges:
- Charge \( q_1 = -4 \, \mu C = -4 \times 10^{-6} \, C \)
- Charge \( q_2 = 8 \, \mu C = 8 \times 10^{-6} \, C \)
The distance between these charges is \( d = 27 \, cm = 0.27 \, m \).

### Step 2: Set Up the Equation
Let the distance from the first charge (negative charge) to the point where the potential is zero be \( x \). Therefore, the distance from the second charge to this point will be \( d - x = 0.27 - x \).

### Step 3: Write the Expression for Electric Potential
The electric potential \( V \) at a point due to a charge is given by:
\[ V = k \frac{q}{r} \]
where \( k = 9 \times 10^9 \, N \cdot m^2/C^2 \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point.

The total potential \( V \) at point \( P \) due to both charges is:
\[ V = V_1 + V_2 = k \frac{-4 \times 10^{-6}}{x} + k \frac{8 \times 10^{-6}}{0.27 - x} \]
Setting this equal to zero for the point where the potential is zero:
\[ k \frac{-4 \times 10^{-6}}{x} + k \frac{8 \times 10^{-6}}{0.27 - x} = 0 \]

### Step 4: Simplify the Equation
Since \( k \) is a constant and appears in both terms, we can cancel it out:
\[ \frac{-4 \times 10^{-6}}{x} + \frac{8 \times 10^{-6}}{0.27 - x} = 0 \]

### Step 5: Rearranging the Equation
Rearranging gives:
\[ \frac{8 \times 10^{-6}}{0.27 - x} = \frac{4 \times 10^{-6}}{x} \]
Cross-multiplying yields:
\[ 8 \times 10^{-6} \cdot x = 4 \times 10^{-6} \cdot (0.27 - x) \]

### Step 6: Solve for \( x \)
Expanding the equation:
\[ 8x = 4(0.27 - x) \]
\[ 8x = 1.08 - 4x \]
Combining like terms:
\[ 8x + 4x = 1.08 \]
\[ 12x = 1.08 \]
\[ x = \frac{1.08}{12} = 0.09 \, m \]

### Step 7: Convert to Centimeters
To convert meters to centimeters:
\[ x = 0.09 \, m = 9 \, cm \]

### Final Answer
The distance from the first charge where the electric potential is zero is **9 cm**.

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