Animated Solution for Physics - Magnetic Effects of Current: A charged particle of mass $m$ and charge $q$ travels on a circular path of radius $r$ that is perpendicular to a magnetic field $B$. The time taken by the particle to complete one revolution is

Have you ever wondered what happens when a tiny, charged particle is shot into a powerful magnetic field? It doesn't just travel in a straight line. Instead, it gets caught in an invisible web, forced to perform a perfect, continuous circular dance. This phenomenon is not just a mathematical curiosity; it is the fundamental principle behind particle accelerators like the cyclotron, which have unlocked the secrets of the atomic nucleus.

In this problem, we are tasked with finding the time it takes for such a particle to complete exactly one revolution. We are given a particle with mass $m$ and charge $q$. It is moving in a circular path of radius $r$. The magnetic field $B$ is uniform and perpendicular to the particle's motion. Let's break down the physics step-by-step and uncover the elegant result hidden within.

Magnetic Lorentz Force

When a charged particle moves through a magnetic field, it experiences a force known as the Magnetic Lorentz Force. The magnitude of this force depends on the charge, the velocity, the magnetic field strength, and the angle between the velocity and the field. The formula is given by $F_m=qvB\sin \theta$.

In our specific scenario, the problem states that the particle's path is perpendicular to the magnetic field. This means the angle $\theta$ is exactly $90^{\circ }$. Since $\sin (90^{\circ })=1$, the expression for the magnetic force simplifies beautifully to $F_m=qvB$.

This force has a very special property: it is always perpendicular to the velocity of the particle. Because it acts at a right angle to the motion, it cannot do any work on the particle. It cannot speed the particle up or slow it down; it can only change its direction.

Dictating the Dance

Before we dive into the math, let's understand why the particle moves in a circle. The direction of the magnetic Lorentz force is given by the Right-Hand Rule. If you point the fingers of your right hand in the direction of the particle's velocity, and then curl them towards the direction of the magnetic field, your thumb points in the direction of the force (for a positive charge).

Because the force is always perpendicular to the velocity, it acts as a steering wheel, constantly turning the particle without ever stepping on the gas or the brakes. This continuous, perpendicular push is the exact recipe for uniform circular motion.

Centripetal Force

Because the magnetic force is constantly pulling the particle sideways, the particle is forced into a circular trajectory. From classical mechanics, we know that any object moving in a circle requires a net force directed towards the center of the circle. This is called the centripetal force.

The formula for centripetal force is $F_c=\frac{m{v}^2}{r}$. Here, $m$ is the mass of the particle, $v$ is its tangential speed, and $r$ is the radius of the circular path. This is not a new, separate force; rather, it is a requirement for circular motion.

In our situation, what is providing this required centripetal pull? It is entirely provided by the magnetic Lorentz force. The magnetic field acts as the invisible string keeping the particle tethered to the center of its circular orbit.

A Balancing Act

Since the magnetic force is the centripetal force, we can set their mathematical expressions equal to each other. This gives us the master equation for this system:

This equation is a treasure trove of information. We can see that there is a velocity term $v$ on both sides of the equation. We can cancel one $v$ from each side, simplifying the relationship to $\frac{mv}{r}=qB$.

By rearranging this equation, we can solve for the radius $r$ of the circular path. Multiplying both sides by $r$ and dividing by $qB$, we get:

This tells us that the radius is directly proportional to the particle's momentum ($mv$) and inversely proportional to the magnetic field strength and charge. A faster or heavier particle will carve out a larger circle!

Calculating the Time Period

Now we arrive at the core question: how long does it take for the particle to complete one full revolution? This duration is known as the time period, denoted by $T$.

In uniform circular motion, the speed $v$ is constant. Therefore, the time taken is simply the total distance traveled divided by the speed. The distance for one complete revolution is the circumference of the circle, which is $2\pi r$. So, our formula for the time period is:

We already found the expression for the radius $r$ in the previous step. Let's substitute $r=\frac{mv}{qB}$ into our time period equation. This yields:

A Beautiful Independence

Look closely at the substituted equation. We have a velocity $v$ in the numerator (from the radius expression) and a velocity $v$ in the denominator. They perfectly cancel each other out!

The final expression for the time period simplifies to:

This is a profound and somewhat counterintuitive result. It reveals that the time it takes for the particle to complete one orbit is completely independent of its velocity or the radius of its path!

Whether the particle is moving slowly in a tight little circle or zipping around in a massive orbit, it will take the exact same amount of time to complete one lap. This beautiful independence is the core operating principle of the cyclotron. It allows scientists to apply an alternating electric field at a constant frequency to continuously accelerate particles to incredibly high energies.