How Magnetic Fields Affect Charged Particle Trajectories
The motion of a charged particle in electric and magnetic fields behaves differently. So, what is the motion of a charged particle in a uniform magnetic field? Today, we will study the motion of a charged particle in a uniform magnetic field. To understand this concept in-depth, we must first understand how does magnetic field lines behave?. We know that both the magnetic and electric forces somehow influence the motion in a magnetic field. Although there will be a change in the trajectory of the particles in both the forces, the charged particle can be an ion or an atom with an electric charge. Below we will learn about the effects of the electric and magnetic force on a charged particle.
Electric Field Vs Magnetic Field Lines
We will use field lines to describe the motion of a charged particle in electric and magnetic fields. So let us start by understanding what these field lines are? There are many conceptual differences between the electric and magnetic field lines. The positively charged particle has an evenly distributed and outward-pointing electric field. For the negative charge, the electric field has a similar structure, but the direction of the field lines is inwards or reverse to that of the positive charge. The field lines create a direct tangent electric field. Also, if the charge density is higher, then the lines are more tightly packed to each other. We can notice that the electric field has no curl. In case both the charges are involved, then positive charges generate field lines, and negative charges terminate them.
As we know, magnets consist of two poles north and south. So the field lines generate from the north pole and terminate at the south pole in the case of magnets. The charges in magnets are always bipolar, i.e. they always come with two poles (north and south) and never exist in a single-pole(monopole). It generates a non-zero curl for the ordinary magnets. If the field lines do not have a perpendicular velocity component, then charged particles move in a spiral fashion around the lines. This concept is widely used to determine the motion of a charged particle in an electric and magnetic field. We can determine the magnetic force exerted by using the right-hand rule. Let us discuss the motion of a charged particle in a magnetic field and motion of a charged particle in a uniform magnetic field.
The Motion of Charged Particle in Electric and Magnetic Field
So, you must be wondering how do we define the motion of a charged particle in a magnetic field and motion of a charged particle in a uniform magnetic field? Suppose if a charged particle is in motion, then the directional component of the force towards motion and the force on the particle performs some amount of work. Let us consider this particle has a charge q and it moves in the direction of magnetic field B (motion in a magnetic field). And the velocity of the particle experiences a perpendicular magnetic force. In this case, the magnetic force does not perform any work on the particle, and hence there is no change in the velocity of the charged particle. Considering the velocity to be v and representing the mathematical equation of this particle perpendicular to the magnetic field where the magnetic force acting on a charged particle of charge q is
F = q(v x B).
Here, the magnetic force becomes centripetal force due to its direction towards the circular motion of the particle. Hence, if the field and velocity are perpendicular to each other, then the particle takes a circular path. Sometimes, the magnetic field and a velocity component are in the same direction. As a result of that, the particle does not experience any effect of the magnetic field, and its magnitude remains the same in the entire motion. The motion resulting from both of these components takes a helical path, as described in the diagram below.
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We can determine the centripetal force perpendicular towards the center while the given radius of the circular path by the particle is r.
\[F = \frac{mv^{2}}{r}\] --(1)
Both magnetic field and velocity experiences perpendicular magnetic force and its magnitude can be determined as follows.
\[F = qvB\] --(2)
Using equation (1) and (2)
\[F = \frac{mv^{2}}{r} = qvB\]
Simplifying the equation above
\[r = \frac{mv}{qB}\]
We know that the angular frequency of the particle is
\[v = \omega r\]
Substituting the value from the above equation in this one,
\[\omega = \frac{qB}{m} = 2\pi v\]
Let us find the time for one revolution(T),
\[T = \frac{2\pi}{\omega} = \frac{1}{v}\]
Total distance moved by the particle in one rotation or pitch can be given as,
\[p = v_{p}T = \frac{2\pi m v_{p}}{qB}\]
Where \[v_{p}\] is the parallel velocity. This process describes how the motion of a charged particle in a magnetic field takes place.
FAQs on Motion of a Charged Particle in Magnetic Field: Concepts & Applications
1. What is the magnetic Lorentz force experienced by a charged particle?
The magnetic Lorentz force is the force experienced by a charged particle when it moves through a magnetic field. This force is dependent on the charge of the particle (q), its velocity (v), and the strength of the magnetic field (B). The formula is given by F = q(v × B). The force is always perpendicular to both the velocity of the particle and the magnetic field, and its direction can be found using Fleming's Left-Hand Rule. This force changes the particle's direction but not its speed.
2. What determines the path of a charged particle in a uniform magnetic field?
The path of a charged particle in a uniform magnetic field is determined by the angle (θ) between the particle's velocity vector (v) and the magnetic field vector (B). There are three main cases:
- Straight Path: If the particle enters parallel or anti-parallel to the magnetic field (θ = 0° or 180°), the magnetic force is zero, and the particle continues to move in a straight line undeflected.
- Circular Path: If the particle enters perpendicular to the magnetic field (θ = 90°), it experiences a constant force that acts as a centripetal force, causing it to move in a uniform circular path.
- Helical Path: If the particle enters at any other angle (0° < θ < 90°), its velocity has components both parallel and perpendicular to the field. The perpendicular component causes circular motion, while the parallel component causes it to drift along the field lines, resulting in a spiral or helical path.
3. Why does a charged particle entering a magnetic field perpendicularly move in a circle?
A charged particle moves in a circle because the magnetic force acting on it is always directed perpendicular to its velocity. According to the principles of motion, a force that is constantly perpendicular to the direction of movement acts as a centripetal force. This force continuously changes the direction of the particle's velocity without altering its speed. Since the particle is constantly being pulled towards a central point while maintaining its speed, it follows a uniform circular path.
4. How can you calculate the radius and time period of a particle's circular path in a magnetic field?
To find the radius and time period, we equate the magnetic Lorentz force to the centripetal force required for circular motion (mv²/r):
- Radius (r): The magnetic force (qvB) provides the centripetal force (mv²/r). By setting qvB = mv²/r, we can solve for the radius, which gives r = mv / qB. This shows the radius is proportional to the particle's momentum (mv).
- Time Period (T): The time period is the time taken to complete one circle (T = 2πr / v). Substituting the formula for r, we get T = 2πm / qB. An important insight here is that the time period is independent of the particle's velocity and the radius of its path.
5. How does the motion of a charged particle differ in a magnetic field compared to an electric field?
The motion differs in several key ways:
- Direction of Force: In an electric field, the force (F = qE) is in the same or opposite direction as the field, regardless of velocity. In a magnetic field, the force (F = q(v × B)) is perpendicular to both the velocity and the field.
- Work and Energy: An electric field can do work on a particle and change its kinetic energy (speed). A magnetic field does no work because the force is always perpendicular to displacement, so it only changes the particle's direction, not its speed or kinetic energy.
- Path: A particle entering an electric field perpendicularly follows a parabolic path. A particle entering a magnetic field perpendicularly follows a circular path.
6. What is a cyclotron and how does it use magnetic fields to accelerate particles?
A cyclotron is a type of particle accelerator used to accelerate charged particles like protons to very high energies. It uses a combination of electric and magnetic fields. A strong, uniform magnetic field is applied perpendicularly to two D-shaped hollow metal chambers (called 'Dees'). This field forces the particles to move in a semi-circular path. A high-frequency electric field is applied across the gap between the Dees, which accelerates the particle each time it crosses the gap. As the particle gains speed, the radius of its circular path increases (r = mv/qB), causing it to spiral outwards until it exits the device at high energy.
7. What is a velocity selector and how does it use both electric and magnetic fields?
A velocity selector is a device that uses crossed electric and magnetic fields to allow only particles of a specific velocity to pass through undeflected. The setup consists of a region where a uniform electric field (E) and a uniform magnetic field (B) are perpendicular to each other, and also perpendicular to the initial velocity of the particles. A charged particle entering this region experiences both an electric force (Fₑ = qE) and a magnetic force (Fₘ = qvB) in opposite directions. For a specific velocity where these two forces perfectly balance (Fₑ = Fₘ), the net force on the particle is zero, and it passes straight through. This condition is met when qE = qvB, which simplifies to v = E / B. Particles that are too fast or too slow are deflected and blocked.
