If the maximum value of accelerating potential provided by a ratio frequency oscillator is 12 kV. The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ...............
[mₚ = 1.67 × 10⁻²⁷ kg, e = 1.6 × 10⁻¹⁹ C, Speed of light = 3 × 10⁸ m/s]

Solution

Physics sir · 3 steps

The proton crosses the gap twice every revolution, so each turn gives it 2eV of energy.

1. Target kinetic energy. The final speed is v = c/6 = 5 × 10⁷ m s⁻¹. This is small enough to use K = 1/2mv² = 1/2 × 1.67 × 10⁻²⁷ × 2.5 × 10¹⁵ = 2.0875 × 10⁻¹² J.
2. Energy per revolution. In one revolution the proton crosses the gap between the dees twice. Each crossing gives it eV, so one revolution gives 2eV = 2 × 1.6 × 10⁻¹⁹ × 12000 = 3.84 × 10⁻¹⁵ J.

   N = (1/2mv²)/2eV = (2.0875 × 10⁻¹²)/(3.84 × 10⁻¹⁵) ≈ 543.6

3. So. The proton needs about 543.6 revolutions. The key takes the whole-number part, so the answer is 543.

The common slip: 1087. Counting only one gap crossing per revolution uses eV instead of 2eV, which gives about 1087, double the right count.

The idea to keep. In a cyclotron, number of revolutions = final kinetic energy divided by 2eV. There are two gap crossings per turn, and each crossing gives eV.

Answer from the NTA final answer key. Solution by PrepWiser.