A proton circulates in a cyclotron, beginning approximately...

A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between dees, the electric potential difference between the dees is $200 \mathrm{~V}$. (a) By how much does its kinetic energy increase with each passage through the gap? (b) What is its kinetic energy as it completes 100 passes through the gap? Let $r_{100}$ be the radius of the proton's circular path as it completes those 100 passes and enters a dee, and let $r_{101}$ be its next radius, as it enters a dee the next time. (c) By what percentage does the radius increase when it changes from $r_{100}$ to $r_{101} ?$ That is, what is $$ \text { percentage increase }=\frac{r_{101}-r_{100}}{r_{100}} 100 \% ? $$

A proton circulates in a cyclotron, beginning approximately at rest at the center. Whenever it passes through the gap between dees, the electric potential difference between the dees is $200 \mathrm{~V}$.
(a) By how much does its kinetic energy increase with each passage through the gap? (b) What is its kinetic energy as it completes 100 passes through the gap? Let $r_{100}$ be the radius of the proton's circular path as it completes those 100 passes and enters a dee, and let $r_{101}$ be its next radius, as it enters a dee the next time. (c) By what percentage does the radius increase when it changes from $r_{100}$ to $r_{101} ?$ That is, what is
$$
\text { percentage increase }=\frac{r_{101}-r_{100}}{r_{100}} 100 \% ?
$$

Key Concepts

Electric Potential Energy Gain

In any accelerator, a charged particle gains kinetic energy equal to the product of its charge and the electric potential difference it traverses. This fundamental concept shows that when the particle passes through a gap with a voltage difference, it receives an energy boost, providing the basis for acceleration in devices such as a cyclotron.

Cyclotron Operation and Discrete Acceleration

A cyclotron accelerates charged particles by repeatedly sending them through a gap where an alternating electric field provides fixed energy increments. Each pass through the gap increases the particle’s kinetic energy by a constant amount, and the overall process is synchronized with the particle’s circular motion in a uniform magnetic field, a condition known as cyclotron resonance.

Kinetic Energy and Circular Motion in a Magnetic Field

When a charged particle moves in a uniform magnetic field, it follows a circular path whose radius is dictated by its momentum, mass, and charge (via the relation r = mv/(qB)). Since kinetic energy is related to the square of the velocity, any increase in kinetic energy directly influences the radius of its path, making this relationship crucial for understanding cyclotron dynamics.

Fractional Changes and Percentage Increase in Radius

In the context of cyclotrons, as the kinetic energy increments lead to changes in the circular path radius, understanding fractional changes becomes essential. The radius is proportional to the square root of the kinetic energy; thus, small increases in kinetic energy translate into fractional (and percentage) increases in radius according to differential relations. This concept enables one to calculate the percentage change in the radius as the particle is accelerated.

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