AIPMT PRELIMS 2004 Physics Potential Difference MCQ Question

To solve the problem of determining the potential difference required to accelerate a charged bullet to a certain speed, we need to utilize the concepts of kinetic energy and electric potential energy.

Given Data:

• Mass of the bullet, $m = 2 \, \text{g} = 0.002 \, \text{kg}$ (since 1 g = 0.001 kg)
• Charge of the bullet, $q = 2 \, \mu\text{C} = 2 \times 10^{-6} \, \text{C}$
• Final speed, $v = 10 \, \text{m/s}$

Step 1: Calculate the Kinetic Energy

When the bullet is accelerated from rest to a speed of $v$, it gains kinetic energy given by the formula:

$$KE = \frac{1}{2} mv^2$$

Substituting the known values:

$$KE = \frac{1}{2} \times 0.002 \, \text{kg} \times (10 \, \text{m/s})^2$$

Calculating this:

$$KE = \frac{1}{2} \times 0.002 \times 100 = 0.1 \, \text{J}$$

Step 2: Relate Kinetic Energy to Electric Potential Energy

The work done on the bullet by the electric field when it moves through a potential difference $V$ is equal to the change in electric potential energy, which can be expressed as:

$$W = qV$$

Setting the work done equal to the kinetic energy gained:

$$qV = KE$$

Substituting $KE$ and $q$:

$$2 \times 10^{-6} \, \text{C} \cdot V = 0.1 \, \text{J}$$

Step 3: Solve for the Potential Difference $V$

Rearranging the equation to solve for $V$:

$$V = \frac{KE}{q} = \frac{0.1 \, \text{J}}{2 \times 10^{-6} \, \text{C}}$$

Calculating $V$:

$$V = \frac{0.1}{2 \times 10^{-6}} = 50,000 \, \text{V} = 50 \, \text{kV}$$

Conclusion:

The potential difference required to accelerate the bullet to a speed of 10 m/s is $50 \, \text{kV}$.

Why the Correct Answer is Right:

The calculations show that the potential difference $V$ needed is indeed $50 \, \text{kV}$, making option A the correct answer.

Why Other Options are Incorrect:

• Option B (5 V): This value is far too low to provide the necessary kinetic energy of 0.1 J.
• Option C (50 V): This is also too low; it would yield only $2 \times 10^{-6} \, \text{C} \times 50 \, \text{V} = 1 \times 10^{-4} \, \text{J} = 0.0001 \, \text{J}$, which is insufficient.
• Option D (4.5 kV): Similar reasoning applies; this value would provide even less energy than option C.

Thus, the comprehensive analysis confirms that the correct potential difference is indeed $50 \, \text{kV}$.