Which graph(s) show(s) the correct distribution of potential (at the points between them) due to two like point charges. [X= distance from the charge on left]

Options

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Answer

To determine the correct distribution of electric potential due to two like point charges, we can follow these steps:

### Step 1: Understand the Concept of Electric Potential
Electric potential (V) due to a point charge is given by the formula:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge. For two like charges, both charges will contribute to the total potential at any point in space.

### Step 2: Analyze the Configuration
Let's consider two positive point charges, \( Q_1 \) and \( Q_2 \), separated by a distance \( d \). We will analyze the potential at a point \( X \) which is at a distance \( x \) from \( Q_1 \) and \( (d - x) \) from \( Q_2 \).

### Step 3: Write the Expression for Total Potential
The total electric potential \( V_{total} \) at point \( X \) is the sum of the potentials due to both charges:
\[ V_{total} = V_1 + V_2 = \frac{kQ_1}{x} + \frac{kQ_2}{d - x} \]
Since both charges are equal, we can simplify this to:
\[ V_{total} = \frac{kQ}{x} + \frac{kQ}{d - x} \]

### Step 4: Analyze the Behavior of the Potential
1. As \( x \) approaches 0 (i.e., very close to \( Q_1 \)), \( V_{total} \) approaches infinity because \( \frac{1}{x} \) becomes very large.
2. As \( x \) approaches \( d \) (i.e., very close to \( Q_2 \)), \( V_{total} \) also approaches infinity because \( \frac{1}{d - x} \) becomes very large.
3. At the midpoint \( x = \frac{d}{2} \), the potential will be at a local maximum because the contributions from both charges are equal and additive.

### Step 5: Sketch the Graph
- The graph of potential \( V \) versus distance \( x \) will show:
- A steep increase as \( x \) approaches 0 from the right.
- A steep increase as \( x \) approaches \( d \) from the left.
- A maximum at the midpoint \( x = \frac{d}{2} \).

### Conclusion
The correct graph will show a curve that rises steeply towards infinity as you approach either charge and has a maximum at the midpoint between the two charges.