Question

A particle with charge \(q\) is at rest when a magnetic field is suddenly turned on. The field points in the \(z\) -direction. What is the direction of the net force acting on the charged particle? a) in the \(x\) -direction b) in the \(y\) -direction c) The net force is zero. d) in the \(z\) -direction

Short answer

a) x-direction b) y-direction c) The net force is zero d) z-direction Answer: c) The net force is zero.

Step-by-step solution

1. Understand the magnetic force on a moving charge

The magnetic force acting on a moving charged particle is given by the equation: \(\boldsymbol{F} = q(\boldsymbol{v} \times \boldsymbol{B})\), where \(\boldsymbol{F}\) is the magnetic force, \(q\) is the charge of the particle, \(\boldsymbol{v}\) is the velocity of the particle, and \(\boldsymbol{B}\) is the magnetic field vector.

2. Determine the velocity of the particle

Since the particle is at rest, its velocity, \(\boldsymbol{v}\), is equal to zero.

3. Calculate the magnetic force acting on the particle

Since the velocity of the particle is zero, the cross product \(\boldsymbol{v} \times \boldsymbol{B}\) is also zero. Therefore, the magnetic force acting on the particle is: \(\boldsymbol{F} = q(\boldsymbol{0} \times \boldsymbol{B}) = 0\).

4. Identify the direction of the net force acting on the charged particle

Since the magnetic force acting on the particle is zero, there is no net force acting on the charged particle. Therefore, the correct answer is c) The net force is zero.

Key concepts

Magnetic Field

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A classic example of a magnetic field is the field created around a magnet, where it manifests as an invisible force capable of attracting ferromagnetic materials like iron.

Magnetic fields are represented by the symbol \( \boldsymbol{B} \) and can be visualized through patterns created by iron filings around a magnet. The strength and direction of the magnetic field at any point in space can be determined by the density and orientation of these filings.

In physics, the direction of the magnetic field is defined as the direction that the north end of a compass needle points. Magnetic fields are essential in many technological applications, from electric motors to magnetic resonance imaging (MRI) machines.

Charge

Charge, often denoted by the symbol \( q \) in physics, refers to a property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel each other, while opposite charges attract.

The unit of electric charge is the coulomb (C), which represents a considerable amount of charge. For instance, the charge on a single electron is approximately \( -1.6 \times 10^{-19} \) C.

Charged particles are influenced by electric and magnetic fields; this allows them to do work, interact with other particles, and be manipulated for use in various technologies, like the accelerators used in particle physics.

Lorentz Force

The Lorentz force is the combined electric and magnetic force on a point charge due to electromagnetic fields. Named after the Dutch physicist Hendrik Lorentz, this fundamental force is described by the equation: \( \boldsymbol{F} = q(\boldsymbol{E} + \boldsymbol{v} \times \boldsymbol{B}) \), where \( \boldsymbol{F} \) is the total force exerted on the charged particle, \( q \) is the charge of the particle, \( \boldsymbol{E} \) is the electric field, \( \boldsymbol{v} \) is the velocity of the particle, and \( \boldsymbol{B} \) is the magnetic field.

According to this principle, a charged particle will experience no magnetic force when it is stationary (\( \boldsymbol{v} = 0 \) ), as the cross product with the velocity in the Lorentz force equation will lead to zero. Therefore, the force felt by a charged particle at rest in a magnetic field is solely due to the electric field (if present).

Cross Product in Magnetic Force

The concept of the cross product is crucial when discussing the magnetic force on a moving charged particle. The magnetic part of the Lorentz force is represented by \( \boldsymbol{v} \times \boldsymbol{B} \), which denotes the cross product of the particle's velocity vector \( \boldsymbol{v} \) and the magnetic field vector \( \boldsymbol{B} \).

This cross product results in a vector that is perpendicular to both \( \boldsymbol{v} \) and \( \boldsymbol{B} \), and its magnitude is given by \( |\boldsymbol{v}| \cdot |\boldsymbol{B}| \cdot \sin(\theta) \), where \( \theta \) is the angle between \( \boldsymbol{v} \) and \( \boldsymbol{B} \). It's important to note that the magnitude of the force is at its maximum when \( \boldsymbol{v} \) and \( \boldsymbol{B} \) are perpendicular (\( \sin(90^\circ) = 1 \) ), and it is zero when they are parallel (\( \sin(0^\circ) = 0 \) ), as seen in the textbook exercise when the particle is at rest.