Chapter 28, Problem 3P

To determine

The accuracy of the proton’s position.

Answer to Problem 3P

The answer is $\text{±1}\text{.3}\text{×}{\text{10}}^{\text{-11}}\text{m}$

Explanation of Solution

Given info: Mass of proton, $\text{m = 1}\text{.67}\times {\text{10}}^{-27}\text{kg}$

Speed of proton, $\text{v = }\left(\text{6}\text{.560 ± 0}\text{.012}\right){\text{×10}}^{\text{5}}\text{m/s}\text{=}\text{0}\text{.024}{\text{×10}}^{\text{5}}\text{m/s}$

Formula used:

The uncertainty in the momentum is $\Delta \text{p = m }\Delta \text{v}$

Where,

  $\Delta v$ is change in velocity.

  $m$ is mass

It is known that $\Delta \text{x }\ge \frac{h}{\Delta p}$

Where,

  $h$ is Planck’s constant, $1.055\times {10}^{-34}J\cdot s$

  $\Delta \text{p}$ is uncertainty in the momentum

  $\Delta x$ is the uncertainty in the position measurement

Calculation:

First calculate the uncertainty in the momentum by using the formula $\Delta \text{p = m }\Delta \text{V}$

  $\text{=}\left(\text{1}\text{.67}{\text{×10}}^{\text{-27}}\text{kg}\right)\left(\text{0}\text{.024}{\text{×10}}^{\text{5}}\text{m/s}\right)$

  $\text{= 4}{\text{.00 ×10}}^{\text{-24}}\text{kg }\cdot \text{ m/s}$

Now, calculate the uncertainty in the proton’s position by using the formula $\Delta \text{x }\ge \frac{h}{\Delta p}$

  $\text{=}\frac{\left(\text{1}{\text{.055×10}}^{\text{-34}}\text{J×s}\right)}{\left(\text{4}{\text{.00 ×10}}^{\text{-24}}\text{kg }\cdot \text{ m/s}\right)}\text{=2}\text{.6}{\text{×10}}^{\text{-11}}\text{m}$

Thus, the accuracy of the position is $\text{±1}\text{.3}\text{×}{\text{10}}^{\text{-11}}\text{m}$ .

Conclusion:

The required value is $\text{±1}\text{.3}\text{×}{\text{10}}^{\text{-11}}\text{m}$ .