Two ions having same mass have charges in the ratio 1 : 2. They are projected normally in a uniform

Q: Why do the ions move in a circular path?

When a charged particle moves perpendicularly to a uniform magnetic field, the magnetic force acts perpendicular to its velocity. This force provides the necessary centripetal force, causing the particle to undergo uniform circular motion.

Q: What if the ions entered at an angle other than 90 degrees?

If the angle is not 90 degrees, the velocity component parallel to the magnetic field causes linear motion, while the perpendicular component causes circular motion. The combination of these two motions results in a helical (spiral) path.

Q: Why did we invert the charge ratio in the formula?

Because the radius is inversely proportional to the charge ($r \propto \frac{1}{q}$). So, when calculating the ratio $\frac{r_1}{r_2}$, it depends on $\frac{q_2}{q_1}$, which is the reciprocal of the given ratio $\frac{q_1}{q_2}$.

Q: Does the mass of the ion affect the radius?

Yes, the radius is directly proportional to the mass ($r \propto m$). Heavier ions will have a larger radius if all other factors (velocity, charge, magnetic field) are constant. In this specific problem, the masses were equal, so it didn't affect the final ratio.

Q: What is the significance of the magnetic field B cancelling out?

Since both ions are projected into the *same* uniform magnetic field, the value of $B$ is identical for both. When we take the ratio of their radii, $B$ appears in both the numerator and denominator and cancels out, simplifying our calculation significantly.

Question

Two ions having same mass have charges in the ratio $1 : 2$. They are projected normally in a uniform magnetic field with their speeds in the ratio $2 : 3$. The ratio of the radii of their circular trajectories is

Options:

(A) $1 : 4$

(B) $4 : 3$

(C) $3 : 1$

(D) $2 : 3$

Sigma Percentile · Accepted Answer

Imagine you are observing a high-energy physics experiment. We have two distinct ions, both possessing the exact same mass, but they differ in their electrical charges and the speeds at which they are traveling. They are both shot perpendicularly into a region containing a uniform magnetic field.

Our goal is to determine the ratio of the radii of the circular paths they will trace out. The problem provides us with three crucial pieces of information: their masses are equal ($m_1 = m_2 = m$), the ratio of their charges is $\frac{q_1}{q_2} = \frac{1}{2}$, and the ratio of their speeds is $\frac{v_1}{v_2} = \frac{2}{3}$.

### The Master Equation

When a charged particle enters a magnetic field perpendicularly, the magnetic force acts as a centripetal force, causing the particle to move in a perfect circle. The radius $r$ of this circular path is governed by a fundamental equation in electromagnetism:

$r = \frac{mv}{qB}$

Here, $m$ is the mass, $v$ is the velocity, $q$ is the charge, and $B$ is the magnetic field strength. This equation tells us that the radius is directly proportional to the particle's momentum ($mv$) and inversely proportional to its charge and the magnetic field strength.

### Setting up the Ratio

Since we need to find the ratio of the radii for the two ions, we can set up an equation by dividing the radius of the first ion by the radius of the second ion:

$\frac{r_1}{r_2} = \frac{\frac{m_1 v_1}{q_1 B}}{\frac{m_2 v_2}{q_2 B}}$

Because both ions are moving through the exact same uniform magnetic field, the value of $B$ is identical for both. This allows us to elegantly cancel out $B$ from our equation, simplifying it to:

$\frac{r_1}{r_2} = \left(\frac{m_1}{m_2}ight) 	imes \left(\frac{v_1}{v_2}ight) 	imes \left(\frac{q_2}{q_1}ight)$

### Final Calculation

Now, we simply substitute the given ratios into our simplified equation. We know the masses are equal, so $\frac{m_1}{m_2} = 1$. The ratio of their speeds is given as $\frac{v_1}{v_2} = \frac{2}{3}$.

**Watch out for the trap here!** The formula requires the ratio $\frac{q_2}{q_1}$, but the problem gives us $\frac{q_1}{q_2} = \frac{1}{2}$. Therefore, we must invert it: $\frac{q_2}{q_1} = \frac{2}{1}$.

Substituting these values yields:

$\frac{r_1}{r_2} = (1) 	imes \left(\frac{2}{3}ight) 	imes \left(\frac{2}{1}ight)$

Multiplying these fractions together, we get:

$\frac{r_1}{r_2} = \frac{4}{3}$

Thus, the ratio of the radii of their circular trajectories is **$4 : 3$**.

Animated Solution for Physics - Magnetic Effects of Current: Two ions having same mass have charges in the ratio $1 : 2$ . They are projected normally in a uniform magnetic field with their speeds in the ratio $2 : 3$ . The ratio of the radii of their circular trajectories is

Select Answer:

Visualized Solution

$Analyzing the Given Data$

$Radius of Circular Path$

$Setting up the Ratio$

$Substituting the Values$

$Final Calculation$

The Sigma Insight: Motion of a Charge in Magnetic Fields

The Master Equation

Setting up the Ratio

Final Calculation

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