1. Calculate the proton's kinetic energy:
* The potential energy gained by the proton as it accelerates through the 500-volt potential difference is converted into kinetic energy.
* Potential energy (PE) = Charge (q) * Voltage (V)
* Kinetic energy (KE) = 1/2 * mass (m) * velocity (v)^2
Since PE = KE, we have:
* q * V = 1/2 * m * v^2
2. Calculate the proton's velocity:
* Rearrange the equation from step 1 to solve for velocity:
* v = √(2 * q * V / m)
* Plug in the values:
* v = √(2 * 1.602 x 10^-19 C * 500 V / 1.672 x 10^-27 kg)
* v ≈ 3.09 x 10^5 m/s
3. Calculate the radius of the circular path:
* The magnetic force on a charged particle moving perpendicular to a magnetic field provides the centripetal force needed for circular motion.
* Magnetic force (F) = q * v * B
* Centripetal force (F) = m * v^2 / r
* Equating these forces: q * v * B = m * v^2 / r
* Rearrange to solve for radius:
* r = (m * v) / (q * B)
4. Plug in the values:
* r = (1.672 x 10^-27 kg * 3.09 x 10^5 m/s) / (1.602 x 10^-19 C * 0.30 T)
* r ≈ 0.011 m or 1.1 cm
Therefore, the radius of the circular path followed by the proton is approximately 1.1 cm.
