Animated Solution for Physics - Magnetic Effects of Current: A beam of protons with speed $4\times 10^5{\text{ ms}}^{-1}$ enters a uniform magnetic field of $0.3\text{ T}$ at an angle of $60^{\circ }$ to the magnetic field. The pitch of the resulting helical path of protons is close to (Take, mass of the proton $=1.67\times 10^{-27}\text{ kg}$ and charge of the proton $=1.69\times 10^{-19}\text{ C}$)

A Proton's Angled Entry

Imagine a proton hurtling through space at a staggering speed of $4\times 10^5{\text{ ms}}^{-1}$. Suddenly, it enters a region with a uniform magnetic field of $0.3\text{ T}$. But here is the catch—it doesn't enter perfectly perpendicular or perfectly parallel. It slices into the field at an angle of $60^{\circ }$.

This specific angle is the secret ingredient that transforms a simple trajectory into a beautiful, complex motion.

The Dual Nature of Motion

Because the proton enters at an angle, its velocity vector splits into two distinct components, each playing a unique role in the proton's journey.

The perpendicular component, $v\sin 60^{\circ }$, interacts with the magnetic field to create a magnetic force. This force acts as a centripetal force, constantly pulling the proton into a circular loop.

Meanwhile, the parallel component, $v\cos 60^{\circ }$, is completely unaffected by the magnetic field. It acts as a relentless forward drive, pushing the proton steadily along the magnetic field lines.

A Dance of Circle and Line

When you combine a continuous circular loop with a steady forward push, what do you get? A helix! The proton traces a path that looks exactly like a coiled spring.

The distance the proton travels forward during the exact time it takes to complete one full circular loop is called the pitch of the helix.

Calculating the Pitch

To find the pitch, we rely on a simple kinematic relationship: distance equals speed multiplied by time.

Here, the speed is our forward-driving parallel velocity, $v\cos \theta$. The time is the period of one full revolution, $T$. The time period of a charged particle in a magnetic field is a classic result:

Multiplying these together gives us our master equation for the pitch:

Crunching the Numbers

Now, we carefully substitute the given values into our master equation. We have the mass of the proton $m=1.67\times 10^{-27}\text{ kg}$, its charge $q=1.69\times 10^{-19}\text{ C}$, the magnetic field $B=0.3\text{ T}$, the velocity $v=4\times 10^5{\text{ ms}}^{-1}$, and the angle $\theta =60^{\circ }$.

We know that $\cos 60^{\circ }=\frac{1}{2}$, which neatly cancels out the $2$ in the numerator. Gathering all the powers of $10$ simplifies the expression significantly:

Evaluating the numerical fraction gives us approximately $41.37$, and the powers of $10$ resolve to $10^{-3}$.

Converting this to centimeters, we get exactly $4\text{ cm}$.

Changing the Angle

This problem beautifully illustrates how initial conditions dictate physical behavior. If the proton had entered at $90^{\circ }$, the parallel velocity would be zero, resulting in a pitch of zero—a perfect circle. If it had entered at $0^{\circ }$, the perpendicular velocity would be zero, resulting in a straight line. Always pay close attention to the angle!