Answer
To find the angles at which a distant antenna will not receive any signal from the radio station, we can use the concept of diffraction. The condition for the first minimum in the diffraction pattern is given by:
\[ \sin(\theta) = \frac{m \lambda}{a} \]
Where:
- \(\theta\) is the angle relative to the original direction of the waves.
- \(m\) is the order of the minimum (for the first minimum, \(m = 1\)).
- \(\lambda\) is the wavelength of the radio waves.
- \(a\) is the width of the gap between the skyscrapers.
(a) To find the horizontal angles at which there is no signal, we need to calculate the angle \(\theta\) for the first minimum. For the first minimum, \(m = 1\).
First, let's calculate the wavelength \(\lambda\) from the frequency \(f\):
\[ \lambda = \frac{c}{f} \]
Where:
- \(c\) is the speed of light (\(3 \times 10^8\) m/s).
- \(f\) is the frequency of the radio waves (\(88.9 \times 10^6\) Hz).
Calculate \(\lambda\):
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{88.9 \times 10^6 \, \text{Hz}} \]
Now, calculate \(\lambda\):
\[ \lambda \approx 3.37 \, \text{m} \]
Given that the skyscrapers are 15.0 m apart (\(a = 15.0 \, \text{m}\)), we can find \(\theta\) for the first minimum:
\[ \sin(\theta) = \frac{(1)(3.37 \, \text{m})}{15.0 \, \text{m}} \]
Now, calculate \(\theta\):
\[ \theta \approx 0.225 \, \text{radians} \]
To find the angle in degrees:
\[ \theta \approx 12.9^\circ \]
So, at an angle of approximately \(12.9^\circ\) relative to the original direction of the waves, a distant antenna will not receive any signal from the station.
(b) To find the intensity at ±5.00 degrees from the center (where the first minimum occurs), we can use the single-slit diffraction formula for intensity:
\[ I(\theta) = I_0 \left(\frac{\sin(\alpha)}{\alpha}\right)^2 \]
Where:
- \(I(\theta)\) is the intensity at angle \(\theta\).
- \(I_0\) is the maximum intensity (given as 3.50 W/m²).
- \(\alpha\) is the angle from the center (\(\alpha = \pi a \sin(\theta) / \lambda\)).
For \(\theta = ±5.00^\circ\), we can calculate \(\alpha\) and then use it to find \(I(\theta)\).
First, convert \(\theta\) to radians:
\[ \theta = ±5.00^\circ \times \left(\frac{\pi}{180^\circ}\right) \]
Now, calculate \(\alpha\):
\[ \alpha = \frac{\pi \cdot 15.0 \, \text{m} \cdot \sin(\theta)}{3.37 \, \text{m}} \]
Now, calculate \(I(\theta)\) using the given values:
\[ I(\theta) = (3.50 \, \text{W/m²}) \left(\frac{\sin(\alpha)}{\alpha}\right)^2 \]
Calculate \(I(\theta)\) for both \(+5.00^\circ\) and \(-5.00^\circ\) to find the intensity at those angles.
Work Step by Step
To find the angles at which a distant antenna will not receive any signal from the radio station, we can use the concept of diffraction. The condition for the first minimum in the diffraction pattern is given by:
\[ \sin(\theta) = \frac{m \lambda}{a} \]
Where:
- \(\theta\) is the angle relative to the original direction of the waves.
- \(m\) is the order of the minimum (for the first minimum, \(m = 1\)).
- \(\lambda\) is the wavelength of the radio waves.
- \(a\) is the width of the gap between the skyscrapers.
(a) To find the horizontal angles at which there is no signal, we need to calculate the angle \(\theta\) for the first minimum. For the first minimum, \(m = 1\).
First, let's calculate the wavelength \(\lambda\) from the frequency \(f\):
\[ \lambda = \frac{c}{f} \]
Where:
- \(c\) is the speed of light (\(3 \times 10^8\) m/s).
- \(f\) is the frequency of the radio waves (\(88.9 \times 10^6\) Hz).
Calculate \(\lambda\):
\[ \lambda = \frac{3 \times 10^8 \, \text{m/s}}{88.9 \times 10^6 \, \text{Hz}} \]
Now, calculate \(\lambda\):
\[ \lambda \approx 3.37 \, \text{m} \]
Given that the skyscrapers are 15.0 m apart (\(a = 15.0 \, \text{m}\)), we can find \(\theta\) for the first minimum:
\[ \sin(\theta) = \frac{(1)(3.37 \, \text{m})}{15.0 \, \text{m}} \]
Now, calculate \(\theta\):
\[ \theta \approx 0.225 \, \text{radians} \]
To find the angle in degrees:
\[ \theta \approx 12.9^\circ \]
So, at an angle of approximately \(12.9^\circ\) relative to the original direction of the waves, a distant antenna will not receive any signal from the station.
(b) To find the intensity at ±5.00 degrees from the center (where the first minimum occurs), we can use the single-slit diffraction formula for intensity:
\[ I(\theta) = I_0 \left(\frac{\sin(\alpha)}{\alpha}\right)^2 \]
Where:
- \(I(\theta)\) is the intensity at angle \(\theta\).
- \(I_0\) is the maximum intensity (given as 3.50 W/m²).
- \(\alpha\) is the angle from the center (\(\alpha = \pi a \sin(\theta) / \lambda\)).
For \(\theta = ±5.00^\circ\), we can calculate \(\alpha\) and then use it to find \(I(\theta)\).
First, convert \(\theta\) to radians:
\[ \theta = ±5.00^\circ \times \left(\frac{\pi}{180^\circ}\right) \]
Now, calculate \(\alpha\):
\[ \alpha = \frac{\pi \cdot 15.0 \, \text{m} \cdot \sin(\theta)}{3.37 \, \text{m}} \]
Now, calculate \(I(\theta)\) using the given values:
\[ I(\theta) = (3.50 \, \text{W/m²}) \left(\frac{\sin(\alpha)}{\alpha}\right)^2 \]
Calculate \(I(\theta)\) for both \(+5.00^\circ\) and \(-5.00^\circ\) to find the intensity at those angles.
