Project Seafarer was an ambitious program to construct an enormous antenna, buried underground on a site about 10000 km2 in area. Its purpose was to transmit signals to submarines while they were deeply submerged. If the effective wavelength were 1.5 x 102 Earth radii, what would be the (a) frequency and (b) period of the radiation emitted? Ordinarily, electromagnetic radiations do not penetrate very far into conductors such as seawater. Take the Earth's radius to be 6370 km.

To calculate the frequency and period of the radiation emitted by Project Seafarer, we'll use the following relationships involving wavelength, frequency, and period:

1. **Speed of light (\(c\))**: Approximately \(3.00 \times 10^8\) m/s.
2. **Wavelength (\(\lambda\))**: Given as \(1.5 \times 10^2\) Earth radii.

### Step 1: Calculate the Wavelength

The Earth's radius is \(6370\) km, so:

\[
\lambda = 1.5 \times 10^2 \times 6370 \text{ km}
\]
\[
\lambda = 1.5 \times 100 \times 6370 \text{ km} = 955500 \text{ km} = 955500 \times 10^3 \text{ m} = 9.555 \times 10^8 \text{ m}
\]

### Step 2: Calculate Frequency (\(f\))

Using the relationship:

\[
f = \frac{c}{\lambda}
\]

Substituting in the values:

\[
f = \frac{3.00 \times 10^8 \text{ m/s}}{9.555 \times 10^8 \text{ m}} \approx 0.314 \text{ Hz}
\]

### Step 3: Calculate Period (\(T\))

The period is the reciprocal of frequency:

\[
T = \frac{1}{f} = \frac{1}{0.314} \approx 3.18 \text{ s}
\]

### Summary of Results

(a) **Frequency**: \(f \approx 0.314 \text{ Hz}\)

(b) **Period**: \(T \approx 3.18 \text{ s}\)

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:

1. What are the implications of this frequency for submarine communication?
2. How does seawater's conductivity affect electromagnetic wave propagation?
3. What other frequencies are typically used in submarine communication?
4. How would the design of antennas change for different wavelengths?
5. What are some practical challenges in constructing such a large antenna? 

**Tip**: Always ensure that the wavelength used in calculations is in the correct unit (meters) for consistency with speed and frequency calculations.

Explore how to calculate the frequency and period of a submarine signal with an effective wavelength of 1.5 x 10^2 Earth radii, used in Project Seafarer. Learn about wave physics and electromagnetic wave equations, applying calculations for wavelength, frequency, and period. Suitable for advanced high school students in Grades 11-12.

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