Physics 9702/11 — October/November 2010

Background Concept

Frequency $f$ is the number of cycles (oscillations) per second. Period $T$ is the time taken for one complete cycle.

They are reciprocals:

$$T = \frac{1}{f}$$

with $f$ in $\text{Hz}$ ($=\text{s}^{-1}$) and $T$ in seconds.

Useful prefixes:

• $1\ \text{MHz} = 10^{6}\ \text{Hz}$
• $1\ \text{ns} = 10^{-9}\ \text{s}$
• $1\ \mu\text{s} = 10^{-6}\ \text{s}$

Understanding the Question

You are given a signal frequency of $2.0\ \text{MHz}$ and asked for its period (time for one cycle). The answer choices are in microseconds ($\mu\text{s}$) and nanoseconds ($\text{ns}$), so after finding $T$ in seconds you should convert to one of these units.

Approach

1. Convert $2.0\ \text{MHz}$ into $\text{Hz}$.
2. Use $T=1/f$ to calculate the period in seconds.
3. Convert seconds into $\text{ns}$ or $\mu\text{s}$ and match to the options.

Step-by-Step Reasoning

Convert frequency to Hz:

$$f = 2.0\ \text{MHz} = 2.0 \times 10^{6}\ \text{Hz}$$

Calculate period:

$$T = \frac{1}{f} = \frac{1}{2.0 \times 10^{6}} = 5.0 \times 10^{-7}\ \text{s}$$

Convert $5.0 \times 10^{-7}\ \text{s}$ to nanoseconds:

$$5.0 \times 10^{-7}\ \text{s} = 5.0 \times 10^{-7} \times 10^{9}\ \text{ns} = 5.0 \times 10^{2}\ \text{ns} = 500\ \text{ns}$$

So the correct option is D.

Key Takeaways

• Period and frequency are reciprocals: $T=1/f$.
• Always convert to SI units (Hz) before substituting.
• Be fluent with prefixes (MHz, ns, $\mu$s) to match answers.

Common Mistakes

• Using $T=f$ or $T=2\pi f$ (confusing angular frequency $\omega=2\pi f$ with frequency).
• Converting MHz incorrectly (e.g. treating $\text{MHz}$ as $10^{-6}$ instead of $10^{6}$).
• Incorrect time conversion: mixing up $\text{ns}$ ($10^{-9}$ s) and $\mu\text{s}$ ($10^{-6}$ s).

Things to Be Careful About

• Keep powers of ten consistent: $\text{MHz}=10^{6}$, so the period should be of order $10^{-6}$ to $10^{-7}$ seconds here.
• Check plausibility: higher frequency means smaller period; $2\ \text{MHz}$ should give a period less than $1\ \mu\text{s}$, so $500\ \text{ns}$ is sensible.