1.3 Estimation of Physical Quantities

Orders of Magnitude

Order of magnitude refers to the power of ten that best approximates the size or value of a quantity. This concept is useful for comparing the scale of different objects or quantities. For example:

• The diameter of a nucleus is approximately $10^{-14}$ metres.
• 100 metres is two orders of magnitude larger than 1 metre, as $10^2$ metres is two powers of ten greater.

Estimating to the Nearest Order of Magnitude

To give an estimate to the nearest order of magnitude, follow these steps:

1. Calculate the actual value or use the value provided.
2. Express the result in scientific notation.
3. Round to the nearest power of ten. For example, if you are asked to estimate the area of a hydrogen atom (considering it spherical), where the diameter is given as metres:

• Use the formula for the area of a circle: $A = \pi r^2$ The radius is half the diameter, so: $r = \frac{1.06 \times 10^{-10}}{2} = 0.53 \times 10^{-10} \text{ m}$
• Now calculate the area: $A = \pi \times (0.53 \times 10^{-10})^2 = 8.82 \times 10^{-21} \text{ m}^2$ This value is close to $10^{-20}$, so the order of magnitude of the area is $10^{-20} \text{ m}^2$.

The Importance of Estimation in Physics

Estimation is a fundamental skill in physics. It allows physicists to make quick, rough calculations to check if a value seems reasonable or to make comparisons between different quantities. Estimation helps identify errors in calculations and provides a sense of scale for physical quantities, especially when exact data is unavailable.

When estimating, focus on the significant figures and ensure the approximation reflects the nearest power of ten. For example, if a calculated value is $9.5 \times 10^3$, this would be rounded to $10^4$ as its nearest order of magnitude.