Problem 23 - Physics

Question

A particle of mass \(6.0 \times 10^{-8} \mathrm{~kg}\) and charge \(+7.2 \mu \mathrm{C}\) is traveling due east. It enters perpendicularly a magnetic field whose magnitude is \(3.0 \mathrm{~T}\). After entering the field, the particle completes one-half of a circle and exits the field traveling due west. How much time does the particle spend traveling in the magnetic field?

Step-by-Step Solution

Answer

The particle spends approximately 8.68 ms in the magnetic field.

Step 1: Understand the Motion in a Magnetic Field

When a charged particle moves in a magnetic field, it experiences a force perpendicular to its velocity and the magnetic field direction. This causes the particle to move in a circular path if it enters the field perpendicular to it. The time to travel in a full circle can be found using the relationship between magnetic force and centripetal force.

Step 2: Find the Centripetal Force

The magnetic force acting as the centripetal force is given by \( F = qvB \), where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength. This force also equals the centripetal force \( F = \frac{mv^2}{r} \), where \( m \) is the mass and \( r \) is the radius of the circular path.

Step 3: Express Velocity in Terms of Other Variables

Set the magnetic force equation equal to the centripetal force equation:\[ qvB = \frac{mv^2}{r} \]Simplify to solve for velocity \( v \) in terms of \( q, B, m \):\[ v = \frac{qBr}{m} \]

Step 4: Find the Radius of Circular Motion

Use the expression for velocity to find the radius \( r \) of the path:\[ r = \frac{mv}{qB} \]Here, \( v \) cancels during simplification since it isn't given directly for this problem.

Step 5: Determine the Half-circle Time

The time for a full circle is given by \( T = \frac{2\pi r}{v} \). For half a circle, it is:\[ T_{half} = \frac{\pi r}{v} \]Substitute the value of \( v \) from Step 3 into this equation to find the time for half a circle:\[ T_{half} = \pi \frac{m}{qB} \]

Step 6: Calculate Time with Given Variables

Substitute the given values:\( q = 7.2 \times 10^{-6} \mathrm{~C} \), \( m = 6.0 \times 10^{-8} \mathrm{~kg} \), and \( B = 3.0 \mathrm{~T} \) into the formula:\[ T_{half} = \pi \frac{6.0 \times 10^{-8}}{7.2 \times 10^{-6} \times 3.0} \]Calculate this to find:\[ T_{half} \approx 8.68 \times 10^{-3} \mathrm{~s} = 8.68 \mathrm{~ms} \]

Key Concepts

Centripetal Force

Centripetal force is a crucial concept in understanding the motion of particles in circular paths. When a particle travels in a circular path, there must be a net force acting towards the center of the circle. This force is known as the centripetal force.
This inward force is necessary to keep the particle moving in a circular trajectory rather than in a straight line.

In the context of motion in magnetic fields, the centripetal force is provided by the magnetic force. The particle's mass (\( m \) ) and velocity (\( v \) ) dictate its angular motion, while the radius (\( r \) ) of the path reflects the balance between the speed and the magnetic influence.
This concept is mathematically represented by \[ F_c = \frac{mv^2}{r} \] , which defines the relationship between these variables. When the magnetic force equals the centripetal force, stable circular motion occurs, allowing for precise calculations of motion dynamics.

Magnetic Force

Magnetic force arises from the interaction between charged particles and magnetic fields. When a charged particle, like an electron or proton, enters a magnetic field at an angle, it experiences a force that is perpendicular to both its velocity and the direction of the magnetic field.
This perpendicular force causes the particle to follow a curved path, often circular in nature if the particle moves perpendicular to the field lines.

The magnitude of this force can be calculated using the equation \( F_B = qvB \) , where \( q \) is the charge of the particle, \( v \) is its velocity, and \( B \) is the strength of the magnetic field.
This force is responsible for maintaining the circular motion and acts as the centripetal force required for the particle to move in a circle. Thus, in scenarios like the one described, the magnetic force plays a central role in determining the motion characteristics of charged particles in magnetic fields.

Circular Motion

Circular motion is exhibited by objects or particles that travel along the circumference of a circle. In physics, understanding circular motion is essential when studying systems where forces cause changes in the direction of motion without altering the speed.
In the context of magnetic fields, charged particles often exhibit circular motion when they enter the field perpendicularly due to the continuous perpendicular force acting on them.

This motion is defined by factors like velocity, radius, and the forces involved, which can all be interconnected using the formula for centripetal force as well as the magnetic force equations:

• The radius (\( r \) ) of the circular path can be found from \( r = \frac{mv}{qB} \) .
• The time for completing one circle is given by \( T = \frac{2\pi r}{v} \) , which is crucial for determining the time spent in the field.

Such knowledge enables precise predictions and control over particle paths in practical applications, such as cyclotrons and other magnetic systems.

Charged Particles

Charged particles, such as electrons and protons, carry either positive or negative electrical charges. Their behavior in fields, particularly magnetic fields, is of great interest in physics due to the unique nature of the forces they experience.
When a charged particle enters a magnetic field, its trajectory is influenced by the Lorentz force, steering it into a curve rather than a straight line, under most conditions forming a circular pattern.

This dynamic can be quantitatively described using the equation illustrating magnetic force acting as centripetal force: \( qvB = \frac{mv^2}{r} \) .

• This equation can be manipulated to express other variables, demonstrating the relationship between charge (\( q \) ), velocity (\( v \) ), and magnetic field strength (\( B \) ).
• It highlights how changes in any of these can adjust the particle's motion, providing insight into designing and interpreting experiments based on charged particle dynamics.

From particle accelerators to understanding cosmic phenomena, the motion of charged particles in magnetic fields is a foundational concept in many areas of science and technology.