Question

Two annihilation γ rays in a PET scan originate at the same point and travel to detectors on either side of the patient. If the point of origin is 9.00 cm closer to one of the detectors, what is the difference in arrival times of the photons? This could be used to give position information, but the time difference is small enough to make it difficult.

a. Δ t =Δ x/c
b. Δ t = c/Δ x
c. Δ t = Δ x²/c
d. Δ t = c/Δ x²

Answer 1

The difference in arrival times of gamma-ray photons in a PET scan, when one point of origin is 9.00 cm closer to one of the detectors, is calculated using the formula Δ t = Δ x/c. The correct option is (a) Δ t = Δ x/c.

Step-by-step explanation:

The question relates to a Positron Emission Tomography (PET) scan, a medical imaging technique that detects gamma-ray photons produced by positron-electron annihilation. In this scenario, we are asked to calculate the difference in arrival times of two gamma-ray photons emitted at the same point but detected at different distances from the point of origin.

Since the gamma-ray photons travel at the speed of light (denoted as c), the time taken for a photon to travel a certain distance x is given by the formula t = x/c. The difference in arrival times (Δ t) for photons traveling different distances (with the difference in distance being Δ x) is calculated as Δ t = Δ x/c. Given that the two rays travel in opposite directions and one ray is detected 9.00 cm closer than the other, the difference in distance is 9.00 cm (or 0.09 meters), and the difference in arrival times can be found using this formula.

Plugging the values into the equation, we get:

Δ t = Δ x/c = 0.09 m / (3.00 x 10^8 m/s)

This yields the correct option (a), which can be interpreted as the difference in arrival times being equal to the difference in distance divided by the speed of light:

Δ t = Δ x/c