Problem 14 - University Physics with Modern Physics

Question

A particle with charge \(6.40 \times 10^{-19} \mathrm{C}\) travels in a circular orbit with radius 4.68 \(\mathrm{mm}\) due to the force exerted on it by a magnetic field with magnitude 1.65 \(\mathrm{T}\) and perpendicular to the orbit. (a) What is the magnitude of the linear momentum \(\vec{p}\) of the partcle? (b) What is the magnitude of the angular momentum \(\overrightarrow{\boldsymbol{L}}\) of the particle?

Step-by-Step Solution

Answer

(a) The linear momentum \( p \) is \( 4.97 \times 10^{-21} \; kg \cdot m/s \). (b) The angular momentum \( L \) is \( 2.32 \times 10^{-23} \; kg \cdot m^2/s \).

Step 1: Understanding the Magnetic Force

Magnetic force provides the centripetal force for the particle moving in a circular path. The magnetic force can be expressed as \( F = qvB \), where \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field strength.

Step 2: Relating Magnetic Force to Centripetal Force

For circular motion, the magnetic force acting as the centripetal force is expressed as \( F = \frac{mv^2}{r} \). We can equate \( qvB = \frac{mv^2}{r} \) to solve for the velocity \( v \) and linear momentum \( p = mv \).

Step 3: Solving for Velocity

From the equation \( qvB = \frac{mv^2}{r} \), we can solve for \( v \) by rearranging it to \( v = \frac{qBr}{m} \). However, we need to express \( m \) in terms of momentum \( p = mv \).

Step 4: Calculating Linear Momentum

Use the rearranged formula \( p = qBr \) for linear momentum. Substitute \( q = 6.40 \times 10^{-19} \; C \), \( B = 1.65 \; T \), and \( r = 0.00468 \; m \) into this equation to find \( p \).

Step 5: Calculating Angular Momentum

Angular momentum \( L \) is given by \( L = mvr \). Since \( p = mv \), we have \( L = pr \). Substitute \( r = 0.00468 \; m \) and the previously calculated \( p \) to find \( L \).

Key Concepts

Linear Momentum

Linear momentum is a fundamental concept in physics and describes the amount of motion a particle has. It is represented by the product of the particle's mass and its velocity. In formula terms, it is denoted as \( p = mv \), where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity of the particle.

- Linear momentum helps us understand how much force is needed to stop an object or how much effect that object would have in a collision.- It's a vector quantity, meaning it has both a magnitude and a direction, aligning with the direction of the velocity.

In the context of the circular motion of a charged particle in a magnetic field, understanding linear momentum becomes crucial. The magnetic force acts continuously perpendicular to the velocity of the particle, causing it to move in a circular path. As a result, the magnetic field doesn't change the speed of the particle but constantly redirects its path. To determine the magnitude of the linear momentum for this case, we use the formula \( p = qBr \), where \( q \) is the charge, \( B \) is the magnetic field strength, and \( r \) is the radius of the circular path. This equation shows how momentum is influenced by magnetic as well as geometric factors, providing a clearer picture of motion dynamics.

Angular Momentum

Angular momentum is the rotational analog of linear momentum. It provides insight into how an object rotates and is given by the formula \( L = mvr \), with \( L \) as the angular momentum, \( m \) as the mass, \( v \) as the tangential velocity, and \( r \) as the radius of the circle.

- It helps us understand rotational dynamics in systems, indicating how quickly and in what manner an object will spin.- Like linear momentum, angular momentum is also a vector, having both magnitude and a direction, which is perpendicular to the plane of rotation.

In the context of a charged particle moving in a circular path due to a magnetic field, angular momentum reveals much about the particle's movement characteristics. The relationship \( L = pr \) emerges due to the fact that linear momentum \( p \) replaces \( mv \) when combined with the radius. This simplification shows that, for circular paths under the influence of central forces like magnetism, angular momentum is contingent upon the radius and linear momentum. It's a perfect way to encapsulate how mass distribution and speed affect rotational behavior in a magnetic field.

Circular Motion

Circular motion is a movement that follows the circumference of a circle and can be found in both everyday phenomena and complex systems. It is a fundamental motion type characterized by a constant distance from a central point, known as the radius.

- In uniform circular motion, the speed remains constant, but the direction of the velocity changes continuously.- The centripetal force is essential to keep the object moving in a circle, preventing it from moving off in a straight line due to inertia.

In our exercise involving a charged particle in a magnetic field, the particle follows a circular path through the action of the magnetic force, which serves as the centripetal force. This means the force due to magnetism is always aimed towards the center of the circle, constantly altering the direction of the velocity of the particle without changing its magnitude. The relationship \( F = qvB \) (for magnetic force), where \( F \) is the force, \( q \) is charge, \( v \) is velocity, and \( B \) is the magnetic field, demonstrates how velocity, charge, and field strength dictate the motion's nature. This snapshot into circular motion under magnetic influence allows a clear understanding of dynamics for particles following magnetic lines in various physics and engineering applications.