Question

Specific charges of two particles \(\mathrm{A}\) and \(\mathrm{B}\) are in the ratio \(2: 3 .\) If the mass ratio, \(m_{\mathrm{A}}: m_{\mathrm{B}}\) is \(2: 3\), then the ratio of their charges, \(e_{\mathrm{A}}: e_{\mathrm{B}}\), is (a) \(1: 1\) (b) \(4: 9\) (c) \(9: 4\) (d) \(2: 3\)

Short answer

The ratio of their charges, \(e_{\mathrm{A}}: e_{\mathrm{B}}\), is 1:1.

Step-by-step solution

Understanding Specific Charge

The specific charge of a particle is defined as the ratio of its charge to its mass. This can be expressed as a formula: specific charge = charge (e) / mass (m). Given that the specific charges of two particles A and B are in the ratio 2:3, we can write this as (e_A/m_A) / (e_B/m_B) = 2/3.

Setting up the Ratio

Assuming the mass ratio of two particles A and B is given as 2:3, we can represent this as m_A/m_B = 2/3. We also know the specific charge ratio which is (e_A/m_A) / (e_B/m_B) = 2/3.

Finding the Charge Ratio

To find the charge ratio (e_A/e_B), we can multiply the specific charge ratio by the mass ratio. In this case, we have (e_A/m_A) / (e_B/m_B) = 2/3 and m_A/m_B = 2/3. Multiplying the two ratios we get (e_A/m_A) * (m_B/e_B) = (2/3) * (3/2) which simplifies to e_A/e_B = 1/1. Therefore, the charge ratio is 1:1.

Key concepts

Charge to Mass Ratio

The concept of charge to mass ratio is fundamental in the field of physics, particularly when discussing subatomic particles, ions, or any charged entities. This ratio, often symbolized as \( \frac{e}{m} \), quantifies how the charge \(\text{(e)}\) of a particle relates to its mass \(\text{(m)}\). Understanding this relationship is key for exploring particle dynamics under the influence of electromagnetic fields.

The significance of knowing the charge to mass ratio lies in its ability to predict the behavior of charged particles in accelerators, cathode ray tubes, and even in the vast cosmos under magnetic and electric fields. For example, by using a known magnetic field, one can determine the velocity of a particle if its charge to mass ratio is known. In our exercise, particle A and particle B have specific charges that form a ratio of 2:3, indicating that for every unit of mass, particle A has twice as much charge as particle B.

When approaching problems involving charge to mass ratio, it's crucial to carefully dissect the information provided. This ratio gives us insightful clues about how the particles will behave differently despite perhaps having similar masses or charges; this implication is profound in separating isotopes or identifying unknown particles in mass spectrometry.

Physics Problem Solving

Physics problem solving is a skill that is developed through practice and application of scientific principles. To successfully solve physics problems, students must not only understand the concepts but also know how to apply them in various situations. It requires a step-by-step approach, breaking down complex problems into solvable parts.

Break Down the Problem

Firstly, identify what is being asked and what information is given. For instance, in our exercise concerning charge to mass ratio, we directly deal with two ratios and their relationship.

Formulate Relationships

After identifying the key pieces of information, it is essential to write out the formulas or principles that relate them. As we did, we expressed specific charge as a ratio and utilized the given ratios to set up our equation.

Perform Calculations

With the relevant equations in hand, perform the necessary calculations. For our problem, it involved multiplying two ratios to find the desired charge ratio.

Review Your Solution

Finally, review your solution to ensure that it makes sense physically and mathematically. A sensible check for our exercise would be confirming that the units on both sides of the equation cancel appropriately, leaving a unitless ratio, which is expected for a comparison of two charges.

Competitive Examinations in Chemistry

Competitive examinations in chemistry play a vital role in assessing a student's comprehension and application of chemical principles. These exams range from high school level competitions, like the Chemistry Olympiad, to entrance tests for prestigious universities and graduate programs.

Students preparing for such examinations must have a firm grasp of concepts across various branches of chemistry, including physical chemistry, which often involves calculations similar to our charge to mass ratio problem. Furthermore, competitive chemistry exams test not just theoretical knowledge but also practical problem-solving skills and the ability to apply concepts in unfamiliar contexts.

For success in these competitive environments, students should practice diverse problem sets, understand the underlying principles, and manage their time effectively during preparation and the exam itself. Many competitive chemistry examinations include questions that integrate knowledge from physics, as seen with charge to mass ratio problems. Therefore, a multidisciplinary approach is highly beneficial, reflecting the interconnected nature of sciences.