Problem 102 - Fundamentals of Physics

Question

A magnetic field forces an electron to move in a circle with radial acceleration \(3.0 \times 10^{14} \mathrm{~m} / \mathrm{s}^{2} .\) (a) What is the speed of the electron if the radius of its circular path is \(15 \mathrm{~cm} ?\) (b) What is the period of the motion?

Step-by-Step Solution

Answer

Speed: \(6.7 \times 10^6 \, \mathrm{m/s}\); Period: \(1.4 \times 10^{-7} \, \mathrm{s}\).

Step 1: Identify Given Values

We are given radial acceleration, \( a = 3.0 \times 10^{14} \ \, \mathrm{m/s}^2 \), and the radius of the circular path, \( r = 15 \, \mathrm{cm} \). Convert the radius to meters: \( r = 0.15 \, \mathrm{m} \).

Step 2: Use the Formula for Radial Acceleration

The formula for radial acceleration in circular motion is \( a = \frac{v^2}{r} \). We can reorganize it to solve for velocity, \( v = \sqrt{a \cdot r} \).

Step 3: Calculate the Speed of the Electron

Substitute the known values into the formula: \[ v = \sqrt{3.0 \times 10^{14} \, \mathrm{m/s}^2 \times 0.15 \, \mathrm{m}} \]. This simplifies to \[ v = \sqrt{4.5 \times 10^{13}} \approx 6.7 \times 10^6 \, \mathrm{m/s} \].

Step 4: Determine the Formula for the Period of Motion

The period \( T \) of a circular motion is calculated by \( T = \frac{2\pi r}{v} \), where \( r \) is the radius and \( v \) is the speed obtained in Step 3.

Step 5: Calculate the Period of the Electron's Motion

Using the values from previous steps: \[ T = \frac{2 \pi \times 0.15 \, \mathrm{m}}{6.7 \times 10^6 \, \mathrm{m/s}} \]. This simplifies to \[ T \approx 1.4 \times 10^{-7} \, \mathrm{s} \].

Key Concepts

Magnetic Field

A magnetic field is an invisible force that can affect the motion of charged particles, like electrons. When an electron enters a magnetic field, it experiences a force perpendicular to both its velocity and the direction of the magnetic field.
This causes the electron to move in a circular path.

• The direction of the magnetic field is determined by the right-hand rule.
• Magnetic fields are measured in units called Tesla (T).

When particles move perpendicularly to a magnetic field, they undergo circular motion, with the magnetic force providing the centripetal force required. Understanding how magnetic fields influence electron motion is key in many applications, from particle accelerators to MRI machines.

Radial Acceleration

Radial acceleration is the acceleration that points towards the center of the circle during circular motion. It's often called centripetal acceleration, necessary to keep an object moving in a circular path.
For electrons in a magnetic field, this acceleration is provided by the magnetic force acting perpendicular to its velocity.

• The formula used is \( a = \frac{v^2}{r} \), where \( v \) is the velocity and \( r \) is the radius of the circle.
• This acceleration keeps the electron moving in a stable circular trajectory.

By solving the formula for \( v \), \( v = \sqrt{a \cdot r} \), we calculate the speed of an electron given the radial acceleration and the circular path's radius.

Electron Motion

Electrons exhibit fascinating motion when subject to magnetic fields. Due to the perpendicular nature of the magnetic force to the electron's velocity, they tend to follow a circular path.

• Electrons, being negatively charged, will have a trajectory influenced by the direction of the magnetic field.
• The magnetic force does no work on the electron and only changes the direction of its velocity, not its speed.

This characteristic motion is fundamental in devices like cyclotrons, which use magnetic fields to accelerate electrons to high speeds.

Period of Motion

The period of motion refers to the time taken for the electron to complete one full circle in its circular path. It provides insights into the dynamics of the particle's motion.

• The period \( T \) is given by the formula \( T = \frac{2\pi r}{v} \), where \( r \) is the radius of the path, and \( v \) is the speed of the electron.
• This formula shows that the period is directly proportional to the radius and inversely proportional to the speed.

In contexts such as the exercise task, knowing the period helps you understand the time scale over which the electron's motion takes place, essential for synchronizing operations in electronics and magnetic systems.