Question

Under the action of a given coulombic force the acceleration of an electron is 2.5 $\times$ 10²² m s⁻². Then the magnitude of the acceleration of a proton under the action of same force is nearly

• A. 1.6 $\times$ 10⁻¹⁹ m s⁻²
• B. 9.1 $\times$ 10³¹ m s⁻²
• C. 1.5 $\times$ 10¹⁹ m s⁻²
• D. 1.6 $\times$ 10²⁷ m s⁻²

Hint:

A proton is roughly 1836 times heavier than an electron ($m_p \approx 1836 \cdot m_e$). You can get the answer quickly by dividing the given electron acceleration directly by 1800: $$\frac{2.5 \times 10^{22}}{1836} \approx 1.36 \times 10^{19}\text{ m s}^{-2}$$

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem explores the relationship between force, mass, and acceleration as defined by Newton's Second Law. While both electrons and protons carry the same magnitude of electric charge ($e$), their masses are significantly different. A proton is much heavier than an electron. Therefore, if the same electrostatic force is applied to both particles, the lighter electron will experience a much higher acceleration than the heavier proton. The acceleration is inversely proportional to the mass.
Step 2: Key Formula or Approach:
1. Newton's Second Law: $F = ma$, which implies $a = F/m$.
2. Since the force $F$ is identical for both: $m_e a_e = m_p a_p$.
3. Rearranging to find the proton's acceleration: $a_p = a_e \left( \frac{m_e}{m_p} \right)$.
4. Masses: $m_e \approx 9.1 \times 10^{-31} \text{ kg}$ and $m_p \approx 1.67 \times 10^{-27} \text{ kg}$.
Step 3: Detailed Explanation:

We are given the acceleration of the electron as $a_e = 2.5 \times 10^{22} \text{ m/s}^2$.

We use the mass ratio to find the proton's acceleration: $a_p = (2.5 \times 10^{22}) \times \left( \frac{9.1 \times 10^{-31}}{1.67 \times 10^{-27}} \right)$.

First, let's simplify the ratio of the coefficients: $2.5 \times (9.1 / 1.67) \approx 2.5 \times 5.45 = 13.625$.

Next, combine the powers of 10: $10^{22} \times 10^{-31} / 10^{-27} = 10^{22 - 31 + 27} = 10^{18}$.

This gives us $a_p \approx 13.625 \times 10^{18} \text{ m/s}^2$.

Converting to standard scientific notation, we get $a_p \approx 1.36 \times 10^{19} \text{ m/s}^2$.

Looking at the options provided, the value $1.5 \times 10^{19}$ is the closest approximation to our calculated result.

Step 4: Final Answer:
The magnitude of the acceleration of the proton is nearly 1.5 $\times$ 10¹⁹ m s⁻².