Two particles A and B of masses m_A and m_B respectively and having the same charge are moving in a

Q: Why do the particles follow a circular path?

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts perpendicular to its velocity, providing a centripetal force that causes it to move in a circle.

Q: What if the charges were different?

If the charges were different, the radius would depend on the ratio of momentum to charge ($p/q$). We wouldn't be able to compare their momenta without knowing the exact ratio of their charges.

Q: Why is the magnetic field represented by circles?

The small circles (often with a dot inside) represent magnetic field lines coming out of the plane of the paper towards you. Crosses would represent the field going into the paper.

Q: Can we determine which particle is heavier?

No, we can only determine that the product of mass and velocity (momentum) is greater for particle A. We cannot isolate the mass or velocity individually from this information.

Q: Does the magnetic field do any work on the particles?

No, the magnetic force is always perpendicular to the velocity of the particle, so it does zero work. The kinetic energy and speed of the particles remain constant.

Question

Two particles $A$ and $B$ of masses $m_A$ and $m_B$ respectively and having the same charge are moving in a plane. A uniform magnetic field exists perpendicular to this plane. The speeds of the particles are $v_A$ and $v_B$, respectively and the trajectories are as shown in the figure. Then

Options:

(A) $m_A v_A < m_B v_B$

(B) $m_A v_A > m_B v_B$

(C) $m_A < m_B$ and $v_A < v_B$

(D) $m_A = m_B$ and $v_A = v_B$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are observing two charged particles, $A$ and $B$, as they shoot into a region filled with a uniform magnetic field. The field lines are pointing straight out of the screen towards you, represented by the small circles in the diagram.

As soon as these particles cross the boundary into the magnetic field, they don't just keep going straight. The magnetic field exerts a force on them that is always perpendicular to their velocity. This constant perpendicular force acts exactly like a centripetal force, forcing the particles to travel in perfect circular arcs.

### The Master Equation

To understand exactly how these particles move, we need to bring in the physics. The magnetic force acting on a moving charge is given by $qvB$. Because this force is responsible for the circular motion, we can equate it to the required centripetal force, which is $\frac{mv^2}{r}$.

Setting them equal gives us our master equation:
$$qvB = \frac{mv^2}{r}$$

If we rearrange this to solve for the radius $r$ of the circular path, we get:
$$r = \frac{mv}{qB}$$

This beautiful little equation tells us everything we need to know. It says that the radius of the path depends on the particle's momentum ($mv$), its charge ($q$), and the strength of the magnetic field ($B$).

### Decoding the Visual Clues

Now, let's look at the specific constraints given in our problem. We are told that both particles have the **same charge** ($q$) and they are moving in the **same uniform magnetic field** ($B$).

Because $q$ and $B$ are constant for both particles, the denominator in our radius equation is identical for both. This means the radius $r$ is directly proportional to the numerator, which is the momentum ($mv$):
$$r \propto mv$$

This is our golden key! It tells us that whichever particle has a larger momentum will trace out a larger circle.

Now, let's turn our attention back to the diagram. Just by looking at it, it is glaringly obvious that the semicircular path of particle $A$ is much wider than the path of particle $B$.

Mathematically, we can write this observation as:
$$r_A > r_B$$

### The Final Verdict

We are now ready to bring it all together. We know that the radius is directly proportional to the momentum ($r \propto mv$), and we know from the visual evidence that $r_A > r_B$.

By substituting the proportionalities, we arrive at our final, undeniable conclusion:
$$m_A v_A > m_B v_B$$

The momentum of particle $A$ is strictly greater than the momentum of particle $B$. This perfectly matches option (b).

It's a brilliant problem that tests your ability to connect a fundamental physics equation with visual geometric data. Always remember to look for what is constant in a problem—it usually points you straight to the solution!

Select Answer:

Visualized Solution

$S e t u p$

$r = \frac{m v}{q B}$

$r \propto m v$

$r_{A} > r_{B}$

$m_{A} v_{A} > m_{B} v_{B}$

$C o n c l u s i o n$

$V a r ia t i o n s$

The Sigma Insight: Motion of a Charge in Magnetic Fields

Analyzing the Setup

The Master Equation

Decoding the Visual Clues

The Final Verdict

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r=qBmv​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r∝mv

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Variations