Two particles of masses \(m_1\) and \(m_2\) having charges \(q_1\) and \(q_2\) respectively are projected with the same velocity in a region of uniform magnetic field \(\vec{B}\) pointing vertically upward. If they describe circular paths as shown in the figure, one may conclude that :
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When tracking particles in magnetic field problems, always remember that the radius of a circular trajectory is directly proportional to the mass-to-charge ratio (\(m/q\)) when the velocity \(v\) and field \(B\) are held constant:
\[
R \propto \frac{m}{q}
\]
A smaller radius directly implies a smaller value of \(\frac{m}{q}\). Therefore, seeing that curve 1 is tighter (\(R_1 < R_2\)), you can immediately write \(\frac{m_1}{q_1} < \frac{m_2}{q_2}\) and cross-multiply to reach the final answer in seconds.
Concept:
When a charged particle moves in a uniform magnetic field with a velocity perpendicular to the field lines, it experiences a magnetic Lorentz force. This force always acts perpendicular to both the velocity vector and the magnetic field vector, providing the necessary centripetal force required for the particle to move in a stable circular trajectory.
The magnitude of the magnetic force ($F_B$) is given by:
\[
F_B = q v B \sin(\theta)
\]
Since the particles are projected perpendicularly to the magnetic field, \(\theta = 90^\circ\), which implies \(\sin(90^\circ) = 1\). Thus, the force is:
\[
F_B = q v B
\]
This magnetic force balances the centripetal force ($F_C$) required to maintain a circle of radius $R$:
\[
F_C = \frac{m v^2}{R}
\]
Equating these two forces yields the fundamental formula for the radius of curvature of a charged particle in a magnetic field:
\[
q v B = \frac{m v^2}{R} \quad \Rightarrow \quad R = \frac{m v}{q B}
\]
By evaluating this relation under constraints of constant parameters, we can readily deduce the relationship between the physical properties of the two particles.
Step 1: Expressing the radius for both particles based on given constraints.
We are given that both particles are projected with the exact same velocity, let this common speed be \(v_1 = v_2 = v\).
Furthermore, they are moving within the same uniform magnetic field region, so the magnetic field intensity is identical for both, let it be \(B\).
Using the general radius formula derived from the balance of forces, we write the explicit radii expressions for particle 1 and particle 2:
For particle 1:
\[
R_1 = \frac{m_1 v}{q_1 B} \quad \cdots (1)
\]
For particle 2:
\[
R_2 = \frac{m_2 v}{q_2 B} \quad \cdots (2)
\]
Step 2: Comparing the trajectories from the given diagram.
By analyzing the circular paths provided in the graphic, we observe the curves described by each mass.
Path 1 corresponds to particle 1 with mass \(m_1\), and Path 2 corresponds to particle 2 with mass \(m_2\).
The curve for path 1 is sharper and tighter, meaning it turns more abruptly, which visually indicates a smaller radius of curvature. Conversely, path 2 is flatter and more extended, indicating a larger radius of curvature.
Therefore, by inspection of the geometric trajectories in the figure:
\[
R_1 < R_2
\]
Step 3: Setting up the algebraic inequality and simplifying.
Substitute the expressions from equation (1) and equation (2) into the geometric radius inequality \(R_1 < R_2\):
\[
\frac{m_1 v}{q_1 B} < \frac{m_2 v}{q_2 B}
\]
Since the velocity \(v\) and the magnetic field magnitude \(B\) are positive scalar constants shared by both systems, we can divide both sides of the inequality by the common non-zero term \(\frac{v}{B}\):
\[
\frac{m_1}{q_1} < \frac{m_2}{q_2}
\]
Step 4: Rearranging into the final requested form.
The question presents the choices as comparisons between the mass ratio \(\frac{m_1}{m_2}\) and the charge ratio.
To isolate \(\frac{m_1}{m_2}\) on the left side of our inequality, we multiply both sides by \(q_1\) (which is positive as both undergo identical directional deflection trends indicating identical sign charges) and divide both sides by \(m_2\):
\[
\frac{m_1}{m_2} < \frac{q_1}{q_2}
\]
This mathematically derived relation perfectly corresponds to Option (C).
