Choose Graph between potential and time for an isolated conductor of finite capacitance C, if its charge varies according to the formula `Q=(alpha t +Q_(0))` coulomb, where `Q_(0)` and `alpha` are positive constant.

Options

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B.
C.
D.

Answer

To solve the problem, we need to analyze the relationship between the charge \( Q \) on an isolated conductor and the potential \( V \) across it, given the formula for charge as \( Q = \alpha t + Q_0 \), where \( Q_0 \) and \( \alpha \) are positive constants.

### Step-by-step Solution:

1. **Understand the relationship between charge and potential:**
The potential \( V \) across a capacitor is given by the formula:
\[ V = \frac{Q}{C} \]
where \( C \) is the capacitance of the conductor.

2. **Substitute the expression for charge into the potential formula:**
Given \( Q = \alpha t + Q_0 \), we can substitute this into the potential formula:
\[ V = \frac{\alpha t + Q_0}{C} \]

3. **Simplify the expression for potential:**
We can rewrite the potential as:
\[ V = \frac{\alpha}{C} t + \frac{Q_0}{C} \]
This is a linear equation in the form \( V = mt + b \), where:
- \( m = \frac{\alpha}{C} \) (the slope)
- \( b = \frac{Q_0}{C} \) (the y-intercept)

4. **Identify the characteristics of the graph:**
- The graph of \( V \) versus \( t \) will be a straight line.
- The slope of the line is positive since both \( \alpha \) and \( C \) are positive constants.
- The y-intercept is also positive since \( Q_0 \) is a positive constant.

5. **Conclusion about the graph:**
Since the slope is constant and positive, and the y-intercept is also a constant positive value, the graph will be a straight line that starts above the origin and rises as time increases.

### Final Answer:
The graph between potential \( V \) and time \( t \) is a straight line with a positive slope, starting from \( \frac{Q_0}{C} \) on the y-axis.

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