Question

A particle with a charge of \(+10.0 \mu \mathrm{C}\) is moving at \(300 \cdot \mathrm{m} / \mathrm{s}\) in the positive \(z\) -direction. a) Find the minimum magnetic field required to keep it moving in a straight line at constant speed if there is a uniform electric field of magnitude \(100 . \mathrm{V} / \mathrm{m}\) pointing in the positive \(y\) -direction. b) Find the minimum magnetic field required to keep the particle moving in a straight line at constant speed if there is a uniform electric field of magnitude \(100 . \mathrm{V} / \mathrm{m}\) pointing in the positive \(z\) -direction.

Short answer

Based on the given step-by-step solution, you can create the following short answer. To find the minimum magnetic field required to keep a charged particle moving in a straight line at a constant speed, we can use the Lorentz force equation and equate the electric and magnetic forces. For an electric field pointing in the positive y-direction, the calculation yields a minimum magnetic field of \(1/3\) T. However, when the electric field is pointing in the positive z-direction, there is no magnetic field required since the force acting on the particle will be zero in that direction, and the minimum magnetic field required is \(0\) T.

Step-by-step solution

Lorentz force equation

Write down the Lorentz force equation for the charged particle under the influence of the electric and magnetic fields: \(F = q(E + vB \times \hat{k})\) The force from the electric field in the y-direction is: \(F_y = qE_y\) The force from the magnetic field is: \(F_B = qvB sin(\theta)\) where \(\theta\) is the angle between the velocity vector and the magnetic field vector.

Equate the electric and magnetic forces

Since the particle needs to move in a straight line at a constant speed, the electric and magnetic forces must balance each other: \(F_y = F_B\) \(→ qE_y = qvB sin(\theta)\)

Solve for the magnetic field

Divide both sides of the equation by \(q\): \(E_y = vB sin(\theta)\) Now, solve for \(B\): \(B = \dfrac{E_y}{v sin(\theta)}\) Put given values in the equation: \(B = \dfrac{100 \text{ V/m}}{300 \text{ m/s} sin(90)}\) \(B = \dfrac{1}{3}\text{ T}\) So the minimum magnetic field required to keep the particle moving in a straight line at a constant speed in part (a) is \(\frac{1}{3}\text{ T}\). b) Electric field pointing in the positive z-direction Notice that when the electric field is in the z-direction, the term \(E + vB \times \hat{k}\) will be in the z-direction, thus F will become zero, and there is no magnetic field needed to balance the electric field and to keep particle's motion in a straight line. So the minimum magnetic field required in part (b) is \(0 \text{ T}\).

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 magnetic field is produced by moving electric charges (such as in a current-carrying wire) and by certain subatomic particles like electrons. Magnetic fields exert forces on other moving charges within them, which is the basis of the operation of electric motors, generators, and many other technological devices.

In physics, the strength and direction of a magnetic field at a particular point are described by a vector. In the here discussed exercise, the magnetic field is applied to balance the force exerted by an electric field on a moving charged particle. The effect of the magnetic field on the charged particle can be quantified using the Lorentz force equation, which is a key concept in understanding charged particle motion in different fields.

Electric Field

An electric field is a region in space around a charged particle or object within which an electric charge would experience force. The electric field acts between two charges in a similar way the gravitational field acts between masses. Electric fields are created around objects that have electric charge, or by a magnetic field that changes with time. The strength of this field can be represented by electric field lines.

The direction of the lines indicates the direction a positive charge would move if placed in the field. In our exercise, a uniform electric field was present. Uniform means that the strength and direction of the field are constant at all points. This simplifies the analysis of charged particle motion, as the force exerted by the field on a charge is the same everywhere in the field.

Charged Particle Motion

When a charged particle such as an electron or proton moves through electric and magnetic fields, it experiences forces due to both fields. The equation that describes the behavior of charged particles under the influence of these fields is the Lorentz force equation. The force on the charged particle due to the electric field is directed along the field lines, while the force due to the magnetic field is perpendicular to both the field lines and the particle's velocity vector.

The equation tells us that the path of a charged particle can be curved by magnetic fields unless the magnetic field is applied in a specific way to counteract the force of the electric field, as in this exercise's scenario. If the forces are balanced, the particle can continue to move in a straight line at a constant speed. This principle is utilized in a variety of applications, including particle accelerators and television tube screens.

Uniform Electric Field

A uniform electric field is characterized by identical strength and the same direction at every point within the field. This results in a constant force on any charge placed in the field. The motion of a charged particle in a uniform electric field without any other forces acting on it will be straight and its acceleration will be constant.

In our exercise, the charged particle was moving in a uniform electric field of magnitude \(100 \text{ V/m}\). When another force, such as one from the magnetic field, is applied perpendicularly, it can balance the electric force. This interplay is crucial in designing and understanding the operation of devices like mass spectrometers and oscilloscopes, which rely on precise control of the motion of charged particles in uniform electric fields.