# Problem Analysis Two radio antennas are separated by \( d = 160 \) m and broadcast **identical** waves of frequency \( = 750 \ MHz. You walk in a **circle of radius 20. m** around their midpoint. The question: **How many maxima (constructive interference points) do you detect as you walk around the circle?** # Step 1: Find the Wavelength The speed of light, \( c = 3.00 \times 10^8 \) m/s. \[ \lambda = \frac{c}{f} \] \[ \lambda = \frac{3.00 \times 10^8}{750 \times 10^6} = .400 \text{ m} \] # Step 2: Path Difference and Maxima Condition For two sources separated by \( d \), as you walk around a **circle** centered at the midpoint, the path difference to any point on the circle is: \[ \Delta r = d \cos\theta \] where \( \theta \) is the angle measured from the perpendicular bisector of the line joining the antennas. Constructive interference (maxima) occurs when: \[ \Delta r = m\lambda, \quad m = , \pm 1, \pm 2, \ldots \] \[ d \cos\theta = m\lambda \] \[ \cos\theta = \frac{m\lambda}{d} \] # Step 3: Possible Values for \( m \) Since \( |\cos\theta| \leq 1 \): \[ -1 \leq \frac{m\lambda}{d} \leq 1 \] \[ |m| \leq \frac{d}{\lambda} \] \[ \frac{d}{\lambda} = \frac{1.60}{.400} = 4 \] So \( m \) can take integer values from \(-4\) to \(+4\). # Step 4: Number of Maxima Around the Circle For each \( m \), there are **two points** on the circle (one for \( +\theta \), one for \( -\theta \)), **except** for \( m = \) (directly on the perpendicular bisector), and the endpoints \( m = \pm 4 \) (where \( \cos\theta = \pm 1 \), i.e., \(\theta = , \pi\)). So, total values of \( m \): \( -4, -3, -2, -1, , 1, 2, 3, 4 \) (9 values). - For \( m = \): 1 maximum (at \( \theta = 90^\circ \) and \( 270^\circ \)) - For each nonzero \( m \): 2 maxima (at \( \pm\theta \)), since \( \cos\theta = \pm x \) gives two angles per \( m \) So total maxima: \[ 1\ (\text{for } m=) + 2 \times 4\ (\text{for } m=1,2,3,4) = 1 + 8 = 9 \] But: **for a full circle, \( m = \pm 4 \) correspond to \( \theta = ^\circ, 180^\circ \), so they are distinct points**. Every \( m \) gives two points except \( m= \). So, total maxima: \[ 2 \times 4\ (\text{for } m = 1,2,3,4) + 1\ (\text{for } m=) = 8 + 1 = 9 \] # **Final Answer** ## Number of Maxima Detected \[ \boxed{9} \] You will detect **9 maxima** as you walk around the circle.