Question

Two particles with masses \(m_{1}\) and \(m_{2}\) and charges \(q\) and \(2 q\) travel with the same velocity, \(v\), and enter a magnetic field of strength \(B\) at the same point, as shown in the figure. In the magnetic field, they move in semicircles with radii \(R\) and \(2 R\). What is the ratio of their masses? Is it possible to apply an electric field that would cause the particles to move in a straight line in the magnetic field? If yes, what would be the magnitude and the direction of the field?

Short answer

In this exercise, we have two particles with different masses and charges moving through a magnetic field. They each travel in a semicircle with different radii. We found that the ratio of their masses is 1:4. To make the particles travel in a straight line, we determined that an electric field should be applied with a magnitude equal to the product of the particles' velocity and the magnetic field (i.e., E = vB) and in the opposite direction to the product of their velocity and magnetic field. By applying this electric field, the Lorentz force acting on the particles will become zero, causing them to move in a straight line.

Step-by-step solution

Find the equation for the Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through an electric and a magnetic field. The equation for the Lorentz force is given by: $$ F_L = q(E + vB \sin \theta) $$ where \(F_L\) is the Lorentz force, \(q\) is the charge of the particle, \(E\) is the electric field, \(v\) is the velocity, \(B\) is the magnetic field, and \(\theta\) is the angle between the velocity and the magnetic field. Since the particles enter the magnetic field at a right angle, we have \(\theta = 90°\), so \(\sin \theta = 1\). Thus, the Lorentz force becomes: $$ F_L = q(E + vB) $$

Relate Lorentz Force to Centripetal Force

In the magnetic field, particles travel in a circular path, experiencing a centripetal force, given by: $$ F_c = \frac{mv^2}{r} $$ where \(F_c\) is the centripetal force, \(m\) is the mass of the particle, \(v\) is the velocity of the particle, and \(r\) is the radius of the circular path. In this case, the Lorentz force is equal to the centripetal force (\(F_L = F_c\)), so we have: $$ q(E + vB) = \frac{mv^2}{r} $$

Solve for the ratio of masses

We have two particles with masses \(m_1\) and \(m_2\), charges \(q\) and \(2q\), velocities \(v\), and radii \(R\) and \(2R\). Applying the relation from step 2 for each particle, we get: $$ m_1 = \frac{qR(E+vB)}{v^2} $$ and $$ m_2 = \frac{2q(2R)(E+vB)}{v^2} $$ To find the ratio of masses, we can divide the expressions for \(m_1\) and \(m_2\): $$ \frac{m_1}{m_2} = \frac{qR(E+vB)}{2q(2R)(E+vB)} = \frac{1}{4} $$ So, the ratio of their masses is \(1:4\).

Determine whether an electric field can make the particles move in a straight line

To make the particles travel in a straight line, the Lorentz force needs to be zero, which means that: $$ F_L = q(E + vB) = 0 $$ If we apply an electric field, \(E = -vB\), then the Lorentz force will become zero and the particles will move in a straight line.

Find the magnitude and direction of the electric field

To make the particles move in a straight line, we need an electric field with the magnitude \(E = vB\), and opposite direction to the product \(vB\).

Key concepts

Centripetal Force

Centripetal force is essential for understanding circular motion. It's the force that keeps an object moving in a curved path and is directed towards the center of the circle around which the object is moving. Imagine you're swinging a ball on a string; the force you exert on the string towards the center of the circle is similar to the centripetal force.

For a charged particle, like those in the exercise, moving with a velocity, \(v\), in a circular path of radius \(r\), the centripetal force is given by the equation \[ F_c = \frac{mv^2}{r} \], where \(m\) is the mass of the particle. In this scenario, the magnetic field provides the centripetal force that makes the charged particles curve in their paths.

Magnetic Field

Imagine a magnetic field as a space where a magnetic force is felt. A simple visual is a field around a magnet where metal objects like paperclips are attracted. This field is quantified by a magnetic field strength, denoted as \(B\).

Magnetic fields interact with moving charged particles, and this interaction can make the particles follow a curved path. The Lorentz force from a magnetic field is always perpendicular to the motion of the charged particle, causing it to move in a circle if there are no other forces acting on the particle. The exercise describes particles entering a uniform magnetic field, prompting a semicircular trajectory due to this force.

Electric Field

An electric field represents the effect of the force that a charge exerts on other charges around it. You can think of it like the ripple effect on water when a stone is thrown in. This field can exert a force on any charged particle placed within it.

In our exercise, the electric field is considered a tool to counterbalance the magnetic force. By adjusting the electric field's strength and direction, we can alter the charged particles' path. Specifically, an electric field of the right magnitude and opposite direction to the velocity times the magnetic field will cancel out the magnetic force, allowing the particles to continue in a straight line.

Charged Particle Motion

When charged particles, such as electrons or ions, move through electric and magnetic fields, they experience the Lorentz force that changes their path. The force's direction and magnitude depend on the particle's charge, velocity, and the angles between its path and the fields.

In the context of the exercise, the charged particles are directed along a semi-circular path due to the magnetic field. However, introducing an electric field can change this motion, leading to a straight-line trajectory. This flexibility in the charged particles' motion provides endless applications, from mundane to complex technologies in various scientific fields.

Radius of Curvature

Radius of curvature pertains to the radius of the circular path a moving object follows. It's a measure of how 'curvy' the path is. If you're drawing circles with a compass, the radius you set for the compass is the radius of curvature for those circles.

In the exercise, the radii of the semicircular paths for the two particles are \(R\) and \(2R\) respectively. This disparity in radii, combined with the known charges and velocity, allows us to derive the ratio of their masses. The radius of curvature is directly connected to the centripetal force required to keep the particle on its circular path and is proportional to the mass of the particles if their speed and the magnetic field strength are constant.