A proton, a deuteron and an alpha-particle are moving with same momentum in a uniform magnetic field

Q: Why is the mass of a deuteron $2m$ and an alpha particle $4m$?

A proton has a mass of approximately 1 amu ($m$). A deuteron is the nucleus of deuterium, containing 1 proton and 1 neutron, so its mass is roughly $1 + 1 = 2$ amu ($2m$). An alpha particle is a helium nucleus with 2 protons and 2 neutrons, giving it a mass of $2 + 2 = 4$ amu ($4m$).

Q: Why did we substitute $v = p/m$ in the force equation?

The problem explicitly states that the particles have the same momentum ($p$). By substituting $v = p/m$ into $F = qvB$, we group all the constant terms ($p$ and $B$) together. This makes it incredibly easy to see that the force only depends on the variable terms, which are $q$ and $m$.

Q: What if the particles had the same kinetic energy instead of momentum?

If kinetic energy ($K$) was constant, we would use the relation $K = \frac{1}{2}mv^2$, which means $v = \sqrt{\frac{2K}{m}}$. The speed ratio would then be proportional to $\frac{1}{\sqrt{m}}$. The force equation would become $F = qB\sqrt{\frac{2K}{m}}$, making the force proportional to $\frac{q}{\sqrt{m}}$.

Q: Why is the magnetic field $B$ ignored when calculating the ratio?

The magnetic field $B$ is uniform, meaning it is exactly the same for all three particles. When you take a ratio (like $F_p : F_d$), the constant $B$ appears in all terms and simply cancels out.

Q: Does the direction of the magnetic field matter for this ratio?

No, the direction of the magnetic field determines the direction of the force (and thus the direction of the circular path), but it does not affect the magnitude of the force or the speed of the particles. The ratios remain the same regardless of the field's orientation, as long as it is perpendicular to the velocity.

Question

A proton, a deuteron and an $\alpha$-particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is ......... and their speed is ......... in the ratio.

Options:

(A) 1 : 2 : 4 and 2 : 1 : 1

(B) 2 : 1 : 1 and 4 : 2 : 1

(C) 4 : 2 : 1 and 2 : 1 : 1

(D) 1 : 2 : 4 and 1 : 1 : 2

Sigma Percentile · Accepted Answer

### The Dance of Charged Particles in a Magnetic Field

Imagine three distinct subatomic particles—a proton, a deuteron, and an alpha particle—entering a uniform magnetic field. They are all moving with the exact same momentum. How do their speeds compare? And how does the magnetic field push them differently? Let's break down this classic physics problem step by step.

### Identifying the Contenders

Before we dive into the math, we must clearly define the properties of our three particles. Let's establish a baseline using the proton:

*   **Proton ($p$):** Mass = $m$, Charge = $e$
*   **Deuteron ($d$):** A deuteron is the nucleus of deuterium (an isotope of hydrogen containing one proton and one neutron). Thus, its mass is roughly twice that of a proton, but its charge remains the same. Mass = $2m$, Charge = $e$
*   **Alpha Particle ($\alpha$):** An alpha particle is a helium nucleus, consisting of two protons and two neutrons. Therefore, its mass is four times that of a proton, and its charge is twice as much. Mass = $4m$, Charge = $2e$

### The Race of Speeds

The problem states that all three particles have the same **momentum ($p$)**. We know from classical mechanics that momentum is the product of mass and velocity:

$$p = mv$$

Rearranging this to solve for velocity, we get:

$$v = \frac{p}{m}$$

Since the momentum $p$ is constant for all three particles, their velocity is inversely proportional to their mass ($v \propto \frac{1}{m}$). Let's calculate the ratio of their speeds:

$$v_p : v_d : v_\alpha = \frac{1}{m_p} : \frac{1}{m_d} : \frac{1}{m_\alpha}$$

Substituting our known mass values:

$$v_p : v_d : v_\alpha = \frac{1}{m} : \frac{1}{2m} : \frac{1}{4m}$$

To simplify this ratio, we multiply the entire expression by $4m$:

$$v_p : v_d : v_\alpha = 4 : 2 : 1$$

This tells us that the lighter proton is zipping along four times faster than the heavy alpha particle to maintain the same momentum!

### The Magnetic Push

Now, let's look at the magnetic force. When a charged particle moves perpendicularly through a magnetic field $B$, it experiences a Lorentz force given by:

$$F = qvB$$

We already found that $v = \frac{p}{m}$. Let's substitute this into our force equation to see how force relates to momentum:

$$F = q\left(\frac{p}{m}\right)B = \left(\frac{q}{m}\right)pB$$

Here is the crucial insight: The momentum $p$ and the magnetic field $B$ are identical for all three particles. Therefore, the magnetic force depends entirely on the ratio of charge to mass, known as the **specific charge** ($\frac{q}{m}$).

Let's calculate the ratio of the magnetic forces:

$$F_p : F_d : F_\alpha = \frac{q_p}{m_p} : \frac{q_d}{m_d} : \frac{q_\alpha}{m_\alpha}$$

Substituting our known values:

$$F_p : F_d : F_\alpha = \frac{e}{m} : \frac{e}{2m} : \frac{2e}{4m}$$

Notice that $\frac{2e}{4m}$ simplifies to $\frac{e}{2m}$. The ratio becomes:

$$F_p : F_d : F_\alpha = 1 : \frac{1}{2} : \frac{1}{2}$$

To clear the fractions, we multiply by 2:

$$F_p : F_d : F_\alpha = 2 : 1 : 1$$

### The Grand Finale

We have successfully decoded the behavior of these particles. The ratio of the magnetic forces acting on them is **$2 : 1 : 1$**, and the ratio of their speeds is **$4 : 2 : 1$**. This elegant interplay between mass, charge, and momentum is a fundamental concept in electromagnetism and a frequent star in competitive exams!

Animated Solution for Physics - Magnetic Effects of Current: A proton, a deuteron and an $α$ -particle are moving with same momentum in a uniform magnetic field. The ratio of magnetic forces acting on them is ......... and their speed is ......... in the ratio.

Select Answer:

Visualized Solution

Properties of Particles

Speed and Momentum Relation

Ratio of Speeds

Magnetic Force Formula

Ratio of Magnetic Forces

Final Conclusion

The Sigma Insight: Motion of a Charge in Magnetic Fields

The Dance of Charged Particles in a Magnetic Field

Identifying the Contenders

The Race of Speeds

The Magnetic Push

The Grand Finale

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