Animated Solution for Physics - Magnetic Effects of Current: If an electron and a proton having same momenta enter perpendicularly to a magnetic field, then

Electron and Proton in a Magnetic Field

Imagine a vast, invisible ocean of magnetic force, directed straight into your screen. Now, picture two tiny dancers entering this stage: an electron and a proton. They are vastly different in mass, yet they enter with the exact same momentum. How will they move? Let's break down the physics behind their elegant trajectories.

When a charged particle enters a uniform magnetic field perpendicularly, it experiences a magnetic force known as the Lorentz force. This force is always perpendicular to the particle's velocity, acting much like the tension in a string swinging a stone. Because the force is perpendicular to the motion, it does no work; it only changes the direction of the particle, forcing it into a perfect circular path.

The necessary centripetal force for this circular motion is provided entirely by the magnetic force. Mathematically, we equate the two:

By rearranging this equation, we can find the radius $r$ of the circular path:

Introducing Momentum

The formula $r=\frac{mv}{qB}$ is incredibly useful, but our problem gives us a specific constraint: the particles have the same momentum.

Recall from classical mechanics that momentum $p$ is simply the product of mass and velocity:

Let's substitute this beautiful, compact variable into our radius equation. The equation transforms into:

This is our master equation for this problem. It tells us exactly what the radius depends on when momentum is the key player.

Now, let's look at our two dancers, the electron and the proton, through the lens of our master equation $r=\frac{p}{qB}$.

1. Momentum ($p$): The problem explicitly states that both particles have the same momentum. So, $p$ is a constant for both. 2. Magnetic Field ($B$): They are entering the same uniform magnetic field, so $B$ is also a constant. 3. Charge ($q$): Here is where it gets interesting. An electron has a charge of $-e$, and a proton has a charge of $+e$. However, the radius of a physical path is a positive distance, so we only care about the magnitude of the charge, $|q|$. For both the electron and the proton, the magnitude of the charge is exactly $e$.

Since $p$, $B$, and $|q|$ are identical for both particles, the radius $r$ must be exactly the same!

Sense of Revolution

While their paths are identical in size, they are not identical in orientation. The magnetic force $\vec{F}=q(\vec{v}\times \vec{B})$ depends heavily on the sign of the charge $q$.

Because the proton is positive and the electron is negative, the magnetic force will push them in exactly opposite directions upon entry. If the proton curves upwards, the electron will curve downwards. They will trace out identical circles, but they will revolve in opposite senses (one clockwise, the other counter-clockwise).

Therefore, ignoring the sense of revolution, the curved path of the electron and the proton will be exactly the same. The beauty of physics lies in how seemingly different particles (one light, one heavy) can be forced into identical geometric paths simply by matching their momentum!