Let's break down the problem step by step: ### **Given:** - Angle θ = 27.7° - Magnetic field strength, B = 3.00 mT = 3.00 × 10⁻³ T - Magnetic force, F = 8.63 × 10⁻¹⁷ N - Charge of proton, q = 1.60 × 10⁻¹⁹ C - Mass of proton, m = 1.67 × 10⁻²⁷ kg --- ## **Step 1: Find the Proton's Speed** The magnetic force on a moving charge is given by: \[ F = q v B \sin\theta \] Solve for \( v \): \[ v = \frac{F}{q B \sin\theta} \] Plug in the values: - \( F = 8.63 \times 10^{-17} \) N - \( q = 1.60 \times 10^{-19} \) C - \( B = 3.00 \times 10^{-3} \) T - \( \sin(27.7^\circ) \approx 0.465 \) \[ v = \frac{8.63 \times 10^{-17}}{(1.60 \times 10^{-19}) \times (3.00 \times 10^{-3}) \times 0.465} \] \[ v \approx \frac{8.63 \times 10^{-17}}{2.232 \times 10^{-22}} \] \[ v \approx 3.87 \times 10^5 \text{ m/s} \] **Explanation:** We used the formula for the magnetic force on a moving charge and solved for speed. --- ## **Step 2: Find Kinetic Energy in Joules** The kinetic energy (\( KE \)) of the proton is: \[ KE = \frac{1}{2} m v^2 \] Substitute the values: - \( m = 1.67 \times 10^{-27} \) kg - \( v = 3.87 \times 10^5 \) m/s \[ KE = \frac{1}{2} \times 1.67 \times 10^{-27} \times (3.87 \times 10^5)^2 \] \[ KE = 0.5 \times 1.67 \times 10^{-27} \times 1.497 \times 10^{11} \] \[ KE = 0.5 \times 1.67 \times 1.497 \times 10^{-16} \] \[ KE \approx 1.25 \times 10^{-16} \text{ J} \] **Explanation:** We used the kinetic energy formula for a non-relativistic particle. --- ## **Step 3: Convert Kinetic Energy to Electron Volts (eV)** 1 eV = \( 1.60 \times 10^{-19} \) J \[ KE_{eV} = \frac{KE}{1.60 \times 10^{-19}} \] \[ KE_{eV} = \frac{1.25 \times 10^{-16}}{1.60 \times 10^{-19}} \] \[ KE_{eV} = 781 \text{ eV} \] **Explanation:** We converted kinetic energy from joules to electron volts using the conversion factor. --- ## **Final Answers:** **(a) Proton's speed:** \( v \approx 3.87 \times 10^5 \) m/s **(b) Kinetic energy:** \( KE \approx 781 \) eV

The concept used involves the magnetic force on a moving charge, which is given by \( F = q v B \sin \theta \). By rearranging this formula, we can determine the particle's speed from the known magnetic force, charge, magnetic field, and angle. Once the speed is found, the kinetic energy is calculated using the classical kinetic energy formula \( KE = \frac{1}{2} m v^2 \). Finally, converting this energy from joules to electron volts allows for a more intuitive understanding of the particle's energy in atomic units.