Problem 83 - Fundamentals of Physics

Question

A particle of mass \(6.0 \mathrm{~g}\) moves at \(4.0 \mathrm{~km} / \mathrm{s}\) in an \(x y\) plane, in a region with a uniform magnetic field given by \(5.0 \hat{\mathrm{i}} \mathrm{mT}\). At one instant, when the particle's velocity is directed \(37^{\circ}\) counterclockwise from the positive direction of the \(x\) axis, the magnetic force on the particle is \(0.48 \hat{\mathrm{k}} \mathrm{N}\). What is the particle's charge?

Step-by-Step Solution

Answer

The particle's charge is 0.04 C.

Step 1: Convert Units

Convert all units to SI units for clarity and consistency in calculations. The particle's mass is given as 6.0 grams, which is equivalent to 0.006 kg. The velocity is 4.0 km/s, which converts to 4000 m/s. The magnetic field is 5.0 mT, which is 5.0 x 10^{-3} T.

Step 2: Understand the Magnetic Force Formula

The magnetic force acting on a charged particle moving in a magnetic field is given by \(\vec{F} = q \vec{v} \times \vec{B} = qvB \sin(\theta)\\), where q is the charge of the particle, v is its velocity, B is the magnetic field, and \( \theta \) is the angle between \( \vec{v} \) and \( \vec{B} \). Here, \( \theta = 37^{\circ} \).

Step 3: Solve for Charge

Rearrange the magnetic force equation to solve for the charge \( q \). The equation becomes \(q = \frac{F}{vB \sin(\theta)}\\). Substitute in \( F = 0.48 \) N, \( v = 4000 \) m/s, \( B = 5.0 \times 10^{-3} \) T and \( \sin(37^{\circ}) \approx 0.60 \).

Step 4: Calculate Charge

Plug in the known values into the rearranged equation: \[ q = \frac{0.48}{4000 \times 5.0 \times 10^{-3} \times 0.60} \]. Simplify the expression: \[ q = \frac{0.48}{12} = 0.04 \]. Thus, the charge of the particle is 0.04 C.

Key Concepts

Charge Calculation

When trying to figure out the charge of a particle, it's essential to understand how it influences the magnetic force experienced by the particle. The charge is a property intrinsic to particles and determines how they interact with electric and magnetic fields.
To calculate the particle's charge in our example, we first need to rearrange the magnetic force equation: \[ F = qvB \sin(\theta) \]Here, \(F\) is the force (in Newtons), \(q\) is the charge (in Coulombs), \(v\) is the velocity (in meters per second), \(B\) is the magnetic field (in Tesla), and \(\theta\) is the angle between the velocity and magnetic field. This formula helps relate these variables when a charged particle moves in a magnetic field.
Rearranging gives us:\[ q = \frac{F}{vB \sin(\theta)} \]Plugging in our values, the charge \(q\) becomes \(0.04\,\text{C}\).
Deeper understanding of charge is vital because it enables us to predict how particles will behave in magnetic fields and conduct further analysis or calculations.

Magnetic Field

Magnetic fields surround magnetic materials and moving electric charges, influencing other nearby charges and magnetic objects. In this exercise, we have a uniform magnetic field characterized by a value of \(5.0 \times 10^{-3}\) T (5.0 mT) directed along the \(\hat{i}\) direction.

• Uniform Magnetic Fields: A uniform magnetic field means the field's strength and direction are consistent throughout the given space.
• Unit Conversion: For calculations, it's crucial to convert measurements to standard units, like converting milliteslas (mT) to teslas (T).

Magnetic fields exert a force on moving charges, dependent on their velocity and the field's strength and orientation. This interaction helps determine whether the charge is deflected, accelerated, or remains (nearly) unaffected.

Particle Motion

The motion of a particle through a magnetic field is fascinating, as it combines kinetic properties and electromagnetic interactions. In our scenario, the particle travels at a speed of \(4.0 \text{ km/s}\) or \(4000 \text{ m/s}\), at an angle of \(37^{\circ}\) from the \(x\)-axis.

• Velocity and Direction: A particle's path and velocity significantly affect how it's influenced by the magnetic field. Here, the particle's angle relative to the \(\hat{i}\) magnetic field is critical, contributing to a sine component in the force equation.
• Calculated Force: To determine the effect on the particle, the magnetic force \(\vec{F}\) relies on speed, direction, and charge, which we've calculated as \(0.48 \text{ N}\) in the \(\hat{k}\) direction.

Overall, particle motion in a magnetic field encapsulates a rich interaction of physics principles, with direction and magnitude playing key roles in the behavior and outcome of the scenario.