A proton moves in a helical path at speed \(v=4.0\) $\times 10^{7} \mathrm{m} / \mathrm{s}$ high above the atmosphere, where Earth's magnetic field has magnitude \(B=1.0 \times 10^{-6} \mathrm{T}\). The proton's velocity makes an angle of \(25^{\circ}\) with the magnetic field. (a) Find the radius of the helix. [Hint: Use the perpendicular component of the velocity.] (b) Find the pitch of the helix-the distance between adjacent "coils." [Hint: Find the time for one revolution; then find how far the proton moves along a field line during that time interval.]

Question: Determine the radius of the helix and the pitch of the helix formed by the proton moving in the magnetic field. Solution: Step 1: Calculate the magnetic force on the proton: \(F = 1.6\times10^{-19}\,\text{C} \times (4.0\times10^7\,\text{m/s} \times \sin(25^\circ)) \times 1.0\times10^{-6}\,\text{T}\). Step 2: Calculate the centripetal force of the helix: \(F_c = \frac{1.67\times10^{-27}\,\text{kg} \times (4.0\times10^7\,\text{m/s} \times \sin(25^\circ))^2}{r}\). Step 3: Equate the magnetic force and centripetal force to find the radius of the helix: \(r = \frac{1.67\times10^{-27}\,\text{kg} \times (4.0\times10^7\,\text{m/s} \times \sin(25^\circ))}{1.6\times10^{-19}\,\text{C} \times 1.0\times10^{-6}\,\text{T}}\). Step 4: Calculate the time for one revolution: \(T = \frac{2\pi r}{4.0\times10^7\,\text{m/s} \times \sin(25^\circ)}\). Step 5: Calculate the pitch of the helix: \(P = (4.0\times10^7\,\text{m/s} \times \cos(25^\circ)) \times T\). Please perform the calculations based on the given formulas and report the numeric values for the radius and the pitch of the helix.