Time Period Calculation for a Sine Wave

Understanding the relationship between frequency and time period is fundamental in electronics and wave theory. A sine wave is a periodic oscillation, and two key characteristics define it: its frequency and its time period.

• Frequency (f): This measures how many complete cycles of the wave occur in one second. It is typically measured in Hertz (Hz).
• Time Period (T): This measures the time it takes for one complete cycle of the wave to occur. It is typically measured in seconds (s).

These two quantities are inversely proportional to each other. The formula that connects them is:

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

Where:

• \(T\) is the time period in seconds.
• \(f\) is the frequency in Hertz.

Converting Frequency for Calculation

The question provides the frequency of the sine wave as 50 kHz. Before we can use the formula \(T = \frac{1}{f}\), we need to convert the frequency from kilohertz (kHz) to Hertz (Hz), as the formula requires frequency in Hz to yield time in seconds.

• Given frequency, \(f = 50 \, \text{kHz}\)
• We know that \(1 \, \text{kHz} = 1000 \, \text{Hz}\).
• Therefore, \(f = 50 \times 1000 \, \text{Hz} = 50000 \, \text{Hz}\).

Calculating the Sine Wave Time Period

Now that we have the frequency in Hertz, we can calculate the time period of the sine wave using the inverse relationship formula:

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

Substitute the value of \(f = 50000 \, \text{Hz}\):

\[T = \frac{1}{50000 \, \text{Hz}}\] \[T = 0.00002 \, \text{seconds}\]

Converting Time Period to Microseconds

The options for the answer are given in microseconds (\(\mu\text{s}\)) and milliseconds (ms). Our calculated time period is \(0.00002\) seconds. To match the options, we need to convert this value to microseconds.

• We know that \(1 \, \text{second} = 10^6 \, \text{microseconds}\) (\(\mu\text{s}\)).
• So, \(T = 0.00002 \, \text{seconds} \times 10^6 \, \frac{\mu\text{s}}{\text{second}}\)
• \(T = 20 \, \mu\text{s}\)

Thus, the time period of a 50 kHz sine wave is 20 microseconds.