Alternating current is called so because the electrons carried by the current move back & forth creating a wave. Alternating direction. Depending on where you are in the world, the number of times per second this wave occurs varies. Each wave, or cycle, back & forth is called a Hertz (Hz).

In New Zealand, our electrical current normally operates at 50Hz.

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Correct answer: 0.01 second

The time for one cycle is the period \(T\), which is related to frequency by:

\[ T = \frac{1}{f} \]

Given:

• \(f = 100\ \mathrm{Hz}\)

Substituting:

\[ T = \frac{1}{100} = 0.01\ \mathrm{s} \]

• 1 second corresponds to 1 Hz.
• 0.0001 second corresponds to 10,000 Hz.
• 10 seconds corresponds to 0.1 Hz.

Therefore, the time for one cycle is 0.01 second.

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Correct answer: D — a cycle is completed 50 times in each second

Frequency is defined as the number of complete cycles that occur per second, measured in hertz (Hz). A "50 hertz current" means the current alternates through one full cycle — rising from zero to a positive peak, back through zero to a negative peak, and returning to zero — exactly 50 times every second. This is the standard AC mains frequency used in New Zealand.

\[ f = \frac{\text{number of cycles}}{\text{time in seconds}} \]

So 50 Hz simply means 50 cycles per second, which can also be written as 50 s⁻¹.

• A is incorrect — voltage (potential difference) is measured in volts; hertz says nothing about voltage magnitude.
• B is incorrect — current magnitude is measured in amperes; hertz describes the rate of alternation, not the quantity of current.
• C is incorrect — power is measured in watts; frequency alone does not determine power dissipation.

Therefore, a 50 Hz current means the waveform completes 50 full cycles every second, which is the definition of frequency in hertz.

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Correct answer: 10 Hz

Frequency is the number of cycles completed per second:

\[ f = \frac{1}{T} \]

where \(T\) is the time for one cycle.

Given:

• \(T = 0.1\ \mathrm{s}\)

Substituting:

\[ f = \frac{1}{0.1} = 10\ \mathrm{Hz} \]

• 1 Hz would require a period of 1 second.
• 100 Hz would require a period of 0.01 second.
• 1000 Hz would require a period of 0.001 second.

Therefore, the frequency is 10 Hz.

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Correct answer: D — a harmonic of the 2 kHz signal

A harmonic is a frequency that is an exact integer multiple of a fundamental frequency. Here, 4 kHz is exactly twice 2 kHz (2 × 2 kHz = 4 kHz), making it the second harmonic of the 2 kHz fundamental. Harmonics appear in impure or non-linear signals as unwanted frequency components at multiples of the original signal frequency.

• A. a fundamental of the 2 kHz signal — The fundamental is the lowest, base frequency of a signal. The 2 kHz component itself is the fundamental; 4 kHz cannot be the fundamental of 2 kHz.
• B. a sub-harmonic of 2 kHz — A sub-harmonic would be a frequency below the fundamental, at an integer fraction of it (e.g., 1 kHz). Since 4 kHz is above 2 kHz, it is not a sub-harmonic.
• C. the DC component of the main signal — A DC component has a frequency of 0 Hz. The 4 kHz signal is clearly an AC frequency component, not DC.

Therefore, the 4 kHz component is a harmonic (specifically the second harmonic) of the 2 kHz fundamental frequency.

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Correct answer: C — hertz

The hertz (symbol Hz) is the SI unit of frequency, defined as exactly one complete cycle per second. It is named after Heinrich Hertz, the German physicist who first demonstrated radio waves. When an alternating signal completes one full oscillation every second, its frequency is 1 Hz.

• Henry is the SI unit of inductance, not frequency.
• Volt is the SI unit of electromotive force (voltage), not frequency.
• Coulomb is the SI unit of electric charge, not frequency.

Therefore, one cycle per second is correctly called one hertz (Hz).

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Correct answer: C — 1000 kHz

The SI prefix "mega" means one million (10⁶). Therefore, one megahertz (1 MHz) equals 1,000,000 Hz. The prefix "kilo" means one thousand (10³), so 1,000,000 Hz divided by 1,000 gives 1,000 kHz.

\[ 1\ \mathrm{MHz} = 1{,}000{,}000\ \mathrm{Hz} = 1{,}000\ \mathrm{kHz} \]

• A — 0.0001 Hz: This is far smaller than 1 MHz; it would represent a very low sub-hertz frequency, not a megahertz value.
• B — 100 kHz: This equals only 0.1 MHz, one tenth of 1 MHz, confusing the kilo and mega prefixes.
• D — 10 Hz: This is an extremely low audio-range frequency, orders of magnitude below 1 MHz.

Therefore, one megahertz is equal to 1000 kHz, following directly from the standard SI prefix definitions.

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Correct answer: D — 1000 MHz

The metric prefix "Giga" (G) means 10⁹, or one billion. A gigahertz (GHz) is therefore one billion hertz (1,000,000,000 Hz). Since one megahertz (MHz) equals one million hertz (10⁶ Hz), dividing gives:

\[ 1\ \text{GHz} = \frac{10^9\ \text{Hz}}{10^6\ \text{Hz/MHz}} = 1000\ \text{MHz} \]

• A — 1000 kHz is incorrect; 1000 kHz equals only 1 MHz, three orders of magnitude too small.
• B — 10 MHz is incorrect; this is just 0.01 GHz.
• C — 100 MHz is incorrect; this equals 0.1 GHz, still a factor of ten too small.

Therefore, one GHz is equal to 1000 MHz, following directly from the standard SI prefix hierarchy kHz → MHz → GHz, each step multiplying by 1000.

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Correct answer: D — 0.707 times the peak voltage

The RMS (Root Mean Square) value represents the equivalent DC voltage that would deliver the same power to a resistive load as the AC waveform. For a sine wave, the RMS value is found by multiplying the peak voltage by 1/√2, which equals approximately 0.707.

\[ V_{\text{rms}} = V_{\text{peak}} \times \frac{1}{\sqrt{2}} = V_{\text{peak}} \times 0.707 \]

Worked example: If a sine wave has a peak voltage of 100 V:

\[ V_{\text{rms}} = 100 \times 0.707 = 70.7\ \mathrm{V} \]

• A. Half the peak voltage (0.5 × peak) — This would be true for a different waveform shape; for a sine wave the RMS value is 0.707 × peak, not 0.5 × peak.
• B. 1.414 times the peak voltage — 1.414 is √2, which is actually the factor used to convert from RMS to peak (i.e. V_peak = V_rms × 1.414), not the other way around.
• C. The peak-to-peak voltage — Peak-to-peak is twice the peak value and is a separate measurement entirely; it is not related to the RMS conversion factor.

Therefore, the RMS value of a sine wave is always 0.707 times its peak voltage, reflecting the effective power-equivalent level of the waveform.

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Correct answer: 7.07 amp

For a sine wave:

\[ I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} \]

Given:

\[ I_{\text{peak}} = 10\ \mathrm{A} \]

So:

\[ I_{\text{rms}} = \frac{10}{\sqrt{2}} \approx 7.07\ \mathrm{A} \]

• 5 A is too low.
• 14.14 A would be peak-to-peak/√2 confusion.
• 20 A is peak-to-peak.

Therefore, the RMS value is 7.07 amp.

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