Question

An unknown particle is caused to move between two electrically charged plates, as illustrated in Figure 2.8. Its path is deflected by a smaller magnitude in the opposite direction from that of a beta particle. What can you conclude about the charge and mass of this unknown particle?

Short answer

The unknown particle has a positive charge and a mass greater than that of the beta particle, as its deflection is smaller in magnitude and in the opposite direction compared to the beta particle.

Step-by-step solution

Understand the Forces Acting on the Charged Particles

The unknown particle and the beta particle both experience a force when moving through the electric field between the charged plates. The force acting on a charged particle in an electric field is given by: F = qE where F is the force, q is the charge of the particle, and E is the electric field strength.

Relate the Force to the Deflection of the Particles

The force acting on the particle will cause it to accelerate, which in turn affects the particle's deflection from its original path. The acceleration can be calculated as: a = F / m where a is the acceleration and m is the mass of the particle. Since the deflection of the unknown particle is smaller and in the opposite direction compared to the beta particle, we can observe that either the charge or mass of the unknown particle differs from that of the beta particle (or both).

Compare the Acceleration of the Unknown Particle With the Beta Particle

The deflection of the unknown particle is smaller in magnitude than the beta particle's; thus, its acceleration is lesser in magnitude. Using the acceleration equation a = F / m, we can now compare the charge-to-mass ratios of both particles: For the unknown particle: \(a_{1} = \frac{q_{1}E}{m_{1}}\) For the beta particle: \(a_{2} = \frac{q_{2}E}{m_{2}}\) Since \(a_{1} < a_{2}\), we can deduce that either \(q_{1} < q_{2}\) and/or \(m_{1} > m_{2}\). We also know that the deflection of the unknown particle is in the opposite direction as the beta particle. This means that the charges of the two particles must be opposite. Using this information, we can make conclusions about the charge and mass of the unknown particle.

Conclusions About the Charge and Mass of the Unknown Particle

Based on our analysis: 1. The charge of the unknown particle is opposite in sign compared to the beta particle. Since the beta particle has a negative charge, the unknown particle has a positive charge. 2. The mass of the unknown particle is greater than the mass of the beta particle. In conclusion, the unknown particle has a positive charge and a mass greater than that of the beta particle.

Key concepts

Charged Particles in Electric Fields

When charged particles enter an electric field, they experience a force that can alter their trajectory. This force, known as the electric force, is fundamental to many processes in physics and is used in a variety of applications, such as in cathode ray tubes and mass spectrometers.

Electric fields are generated between two oppositely charged plates, and their strength is expressed in volts per meter (V/m). When a charged particle, such as an electron or a proton, passes through this field, it will be either attracted to or repelled from the plates depending on its charge. The electric field applies the force symmetrically along the field lines. For example, electrons being negatively charged are drawn toward the positively charged plate, causing a characteristic curve in their path.

Force on Charged Particles

The force acting on a charged particle in an electric field can be calculated using the equation
\( F = qE \)
, where
\( F \) is the force in newtons (N),
\( q \) is the charge of the particle in coulombs (C), and
\( E \) is the electric field strength in volts per meter (V/m). This force is responsible for the acceleration of the particle perpendicular to the field lines.

For example, the force experienced by an electron moving through an electric field will accelerate the electron, which can be detected as a deflection from its original path. Similarly, charged particles used in television screens are directed by controlling the electric fields to create an image.

Mass-to-Charge Ratio

The mass-to-charge ratio (m/q) is a crucial value in understanding the behavior of charged particles in electric fields. It is particularly important in the field of mass spectrometry, where it helps identify the composition of unknown substances.

The mass-to-charge ratio determines how a charged particle will respond when subjected to an electric field. A particle with a high mass-to-charge ratio will experience less acceleration than a particle with a low mass-to-charge ratio, assuming the strength of the electric field is constant. As such, a heavier particle or a particle with less charge will be deflected less than a light particle with the same charge, or a particle with a higher charge of the same mass.

Particle Acceleration in Electric Fields

Particle acceleration in electric fields is the result of the force exerted on charged particles, as described by the equation
\( a = \frac{F}{m} \)
, where
\( a \) is the acceleration,
\( F \) is the force, and
\( m \) is the mass of the particle. The acceleration of a particle directly affects how quickly its velocity changes as it moves through the electric field.

In accelerators, this principle is used to increase the kinetic energy of charged particles to high speeds. The famous Large Hadron Collider (LHC) contains strong electric fields that accelerate particles close to the speed of light. This principle is also at work in an exercise like ours, where understanding acceleration helps us deduce information about the unknown particle's charge and mass based on its deflection pattern.