Motion of Charges in Magnetic Fields

Motion In A Magnetic Field (Helical Motion, Circular Motion)

When a charged particle moves through a region containing only a magnetic field, the force acting on it is given by the magnetic component of the Lorentz force:

$ \vec{F}_B = q (\vec{v} \times \vec{B}) $

where $q$ is the charge of the particle, $\vec{v}$ is its velocity, and $\vec{B}$ is the magnetic field vector.

A crucial aspect of this force is that it is always perpendicular to the velocity vector $\vec{v}$. Mathematically, the cross product $\vec{v} \times \vec{B}$ is always perpendicular to $\vec{v}$.

Since the force is perpendicular to the direction of motion, the magnetic force does no work on the charged particle ($W = \int \vec{F}_B \cdot d\vec{l} = \int \vec{F}_B \cdot \vec{v} dt = 0$, as $\vec{F}_B \perp \vec{v}$).

Work-energy theorem states that the net work done on a particle equals the change in its kinetic energy. Since the magnetic force does no work, the kinetic energy ($KE = \frac{1}{2}mv^2$) of the charged particle remains constant. Consequently, the speed ($v$) of the particle does not change in a magnetic field. The magnetic force only changes the direction of the velocity.

The path of the charged particle in a uniform magnetic field depends on the initial velocity vector $\vec{v}$ relative to the magnetic field vector $\vec{B}$.

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Case 1: Velocity is Parallel or Anti-parallel to the Magnetic Field

If the velocity $\vec{v}$ is parallel ($\theta = 0^\circ$) or anti-parallel ($\theta = 180^\circ$) to the magnetic field $\vec{B}$, the angle between $\vec{v}$ and $\vec{B}$ is $0^\circ$ or $180^\circ$.

The magnetic force magnitude is $F_B = |q| v B \sin\theta$.

Since $\sin 0^\circ = 0$ and $\sin 180^\circ = 0$, the magnetic force $F_B = 0$ in both cases.

If no other force is acting, the particle will continue to move with constant velocity along a straight line, parallel or anti-parallel to the magnetic field lines.

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Case 2: Velocity is Perpendicular to the Magnetic Field (Circular Motion)

If the velocity $\vec{v}$ is perpendicular to the magnetic field $\vec{B}$ ($\theta = 90^\circ$), the magnetic force magnitude is $F_B = |q| v B \sin 90^\circ = |q| v B$.

The direction of this force is perpendicular to both $\vec{v}$ and $\vec{B}$ at all times. Since the magnitude of the force is constant (as $q, v, B$ are constant) and its direction is always perpendicular to the velocity, this force acts as a centripetal force, causing the particle to move in a circular path.

Circular motion of a positive charge in a uniform magnetic field (field into the page).

Let $m$ be the mass of the charged particle and $R$ be the radius of the circular path. The centripetal force required for circular motion is $F_c = \frac{mv^2}{R}$.

The magnetic force provides this centripetal force:

$ F_B = F_c $

$ |q| v B = \frac{mv^2}{R} $

From this equation, we can derive the radius of the circular path:

$ R = \frac{mv^2}{|q| v B} = \frac{mv}{|q|B} $

The radius of the circular path is directly proportional to the momentum ($mv$) of the particle and inversely proportional to the charge magnitude and the magnetic field strength.

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Time Period and Frequency of Circular Motion

The time period ($T$) of the circular motion is the time taken to complete one full circle. It is given by the circumference divided by the speed:

$ T = \frac{2\pi R}{v} $

Substitute the expression for $R$:

$ T = \frac{2\pi (mv/|q|B)}{v} = \frac{2\pi m}{|q|B} $

The frequency ($\nu$) is the number of revolutions per unit time, which is the reciprocal of the time period:

$ \nu = \frac{1}{T} = \frac{|q|B}{2\pi m} $

This frequency is called the cyclotron frequency, as it is a fundamental concept in the working of a cyclotron accelerator (discussed in Section I2).

The angular frequency ($\omega$) is $\omega = 2\pi\nu$:

$ \omega = \frac{|q|B}{m} $

Notice that the time period, frequency, and angular frequency of the circular motion are independent of the speed ($v$) and the radius ($R$) of the orbit. They depend only on the charge-to-mass ratio ($|q|/m$) of the particle and the magnetic field strength ($B$). This property is crucial for the operation of the cyclotron.

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Case 3: Velocity has Components Parallel and Perpendicular to the Magnetic Field (Helical Motion)

If the initial velocity $\vec{v}$ is not parallel or anti-parallel to $\vec{B}$, nor strictly perpendicular, we can resolve $\vec{v}$ into two components:

• $\vec{v}_\parallel$: Component parallel to $\vec{B}$.
• $\vec{v}_\perp$: Component perpendicular to $\vec{B}$.

Let $\theta$ be the angle between $\vec{v}$ and $\vec{B}$. Then $v_\parallel = v \cos\theta$ and $v_\perp = v \sin\theta$.

The magnetic force can be written as the vector sum of forces due to these two components:

$ \vec{F}_B = q (\vec{v} \times \vec{B}) = q [(\vec{v}_\parallel + \vec{v}_\perp) \times \vec{B}] = q (\vec{v}_\parallel \times \vec{B}) + q (\vec{v}_\perp \times \vec{B}) $

• The force component $q(\vec{v}_\parallel \times \vec{B}) = 0$, because $\vec{v}_\parallel$ is parallel to $\vec{B}$. This means the component of velocity parallel to the field, $v_\parallel$, remains constant. The particle moves with uniform velocity along the direction of $\vec{B}$.
• The force component $q(\vec{v}_\perp \times \vec{B})$ has magnitude $|q|v_\perp B$, and its direction is perpendicular to both $\vec{v}_\perp$ and $\vec{B}$. This force causes circular motion in the plane perpendicular to $\vec{B}$ with speed $v_\perp$.

The radius of this circular path is given by the force providing the centripetal acceleration for the perpendicular motion: $|q|v_\perp B = \frac{mv_\perp^2}{R}$.

$ R = \frac{mv_\perp}{|q|B} = \frac{mv\sin\theta}{|q|B} $

The time period of this circular motion is still $T = \frac{2\pi R}{v_\perp} = \frac{2\pi m}{|q|B}$, which is independent of both $v$ and $\theta$.

The combined motion of uniform velocity $v_\parallel$ along the field direction and circular motion with velocity $v_\perp$ in the plane perpendicular to the field is a helical path (a spiral). The axis of the helix is parallel to the magnetic field lines.

Helical motion of a charged particle in a uniform magnetic field.

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Pitch of the Helix

The pitch ($p$) of the helix is the linear distance moved by the particle parallel to the magnetic field direction in one full revolution (one time period $T$).

$ \text{Pitch} = v_\parallel \times T $

Substitute $v_\parallel = v\cos\theta$ and $T = \frac{2\pi m}{|q|B}$:

$ p = (v\cos\theta) \frac{2\pi m}{|q|B} $

If $\theta = 90^\circ$, $v_\parallel = 0$, pitch $= 0$, and the motion is purely circular (Case 2). If $\theta = 0^\circ$ or $180^\circ$, $v_\perp = 0$, $R=0$, and the motion is purely linear (Case 1, a helix with zero radius).

This helical motion is observed in various physical phenomena, such as the motion of charged particles trapped in the Earth's magnetic field (leading to aurora) or in plasma confinement devices.

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Answer:

Motion In Combined Electric And Magnetic Fields

When a charged particle moves in a region where both electric field ($\vec{E}$) and magnetic field ($\vec{B}$) are present, the total force acting on it is the Lorentz force:

$ \vec{F} = \vec{F}_E + \vec{F}_B = q\vec{E} + q(\vec{v} \times \vec{B}) = q(\vec{E} + \vec{v} \times \vec{B}) $

The resulting motion depends on the magnitudes and relative directions of $\vec{E}$, $\vec{B}$, and the initial velocity $\vec{v}$. This combined force allows for the control and manipulation of charged particles.

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Velocity Selector

A velocity selector is a device that uses perpendicular electric and magnetic fields to select charged particles moving at a specific velocity from a beam containing particles with a range of velocities. It's based on the principle that for a particular velocity, the electric force and the magnetic force on a charged particle become equal in magnitude and opposite in direction, resulting in zero net force.

Velocity Selector setup. Perpendicular electric field (down) and magnetic field (into the page).

Setup: Consider a region where a uniform electric field $\vec{E}$ and a uniform magnetic field $\vec{B}$ are perpendicular to each other and also perpendicular to the direction of motion of the charged particles. Let's assume particles are moving along the +x direction, $\vec{v} = v\hat{i}$. Let the electric field be along the +y direction, $\vec{E} = E\hat{j}$, and the magnetic field be along the +z direction, $\vec{B} = B\hat{k}$.

For a positively charged particle ($q>0$) moving with velocity $\vec{v}$, the forces are:

• Electric force: $\vec{F}_E = q\vec{E} = qE\hat{j}$ (along +y).
• Magnetic force: $\vec{F}_B = q(\vec{v} \times \vec{B}) = q(v\hat{i} \times B\hat{k}) = qvB(\hat{i} \times \hat{k})$. Since $\hat{i} \times \hat{k} = -\hat{j}$, $\vec{F}_B = -qvB\hat{j}$ (along -y).

The electric force is in the +y direction, and the magnetic force is in the -y direction. They oppose each other.

The net force on the particle is $\vec{F} = \vec{F}_E + \vec{F}_B = (qE - qvB)\hat{j}$.

For the particle to pass through the region undeflected, the net force must be zero ($\vec{F} = 0$).

$ qE - qvB = 0 $

$ qE = qvB $

Assuming $q \neq 0$, we can divide by $q$:

$ E = vB $

This gives the velocity component along the original direction:

$ v = \frac{E}{B} $

Particles entering this region with velocity $v = E/B$ will experience zero net force and pass straight through. Particles moving slower ($v < E/B$) will experience a stronger electric force ($qE$) than the magnetic force ($qvB$) and will be deflected in the direction of $\vec{E}$. Particles moving faster ($v > E/B$) will experience a stronger magnetic force ($qvB$) and will be deflected in the direction of $\vec{v} \times \vec{B}$ (opposite to $\vec{E}$ for positive charges in this setup).

By placing a slit at the end of the region, only those particles with velocity $v = E/B$ are allowed to pass. This arrangement effectively "selects" particles of a particular velocity. This principle is used in mass spectrometers and electron microscopes.

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Cyclotron

The cyclotron is a particle accelerator invented by Ernest O. Lawrence and M. Stanley Livingston in 1934. It is used to accelerate charged particles (like protons, deuterons, alpha particles, etc.) to high energies for research purposes, medical applications (like producing isotopes), and material science studies.

Diagram of a Cyclotron. Particles are accelerated in the gap between the dees.

Principle: The cyclotron is based on the fact that the time period of revolution of a charged particle in a uniform magnetic field is independent of its speed and the radius of its orbit ($T = \frac{2\pi m}{|q|B}$).

Construction:

• It consists of two large, hollow, semi-circular D-shaped metal containers, called dees, placed horizontally with a small gap between them.
• A uniform magnetic field $\vec{B}$ is applied perpendicular to the plane of the dees (usually provided by a large electromagnet).
• An alternating electric field is applied across the gap between the dees using a high-frequency oscillator.
• A source of charged particles is located near the center of the gap.
• An exit port is provided at the periphery to extract the high-energy particles.

Working:

1. A charged particle is injected near the center of the gap between the dees.
2. The magnetic field forces the particle to move in a semi-circular path inside one of the dees. Inside the dees, there is no electric field (they are hollow metal cavities, acting as Faraday cages).
3. When the particle reaches the gap between the dees, it is subjected to the electric field. The frequency of the alternating electric field is precisely tuned such that the electric field is in a direction that accelerates the particle across the gap. This increases the particle's kinetic energy and speed.
4. With the increased speed, the particle enters the other dee and moves in a larger semi-circular path (since radius $R = mv/|q|B$ increases with $v$).
5. By the time the particle completes its semi-circular path in the second dee and returns to the gap, the alternating electric field reverses its polarity. Thus, the particle is again accelerated across the gap, gaining more energy and speed.
6. This process of acceleration across the gap and movement in increasingly larger semi-circular paths inside the dees is repeated many times. The particle traces a spiral path outwards from the center.
7. When the particle reaches the maximum radius ($R_{max}$) of the dees, it has attained its maximum speed ($v_{max}$) and kinetic energy. It is then deflected by a deflector plate and emerges from the exit port.

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Cyclotron Frequency (Resonance Condition)

The key to the cyclotron's operation is the synchronisation between the particle's revolution period and the alternating electric field. The time taken for a particle to complete one semi-circle inside a dee is half of the period of circular motion, $T/2 = \frac{\pi m}{|q|B}$. This time is independent of the particle's speed and radius.

The frequency of the alternating electric field ($f_{ac}$) must be equal to the cyclotron frequency ($\nu_c$) of the particle, so that the electric field polarity reverses exactly when the particle arrives at the gap. This ensures that the particle is accelerated every time it crosses the gap. This condition ($f_{ac} = \nu_c$) is called the resonance condition.

$ f_{ac} = \nu_c = \frac{|q|B}{2\pi m} $

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Maximum Kinetic Energy of the Accelerated Particle

The particles are accelerated until they reach the outer edge of the dees, where the radius is $R_{max}$. At this maximum radius, the particle has its maximum speed $v_{max}$. The radius formula is $R = \frac{mv}{|q|B}$, so at maximum radius:

$ R_{max} = \frac{m v_{max}}{|q|B} $

$ v_{max} = \frac{|q|B R_{max}}{m} $

The maximum kinetic energy ($K_{max}$) achieved by the particle is:

$ K_{max} = \frac{1}{2} m v_{max}^2 = \frac{1}{2} m \left(\frac{|q|B R_{max}}{m}\right)^2 $

$ K_{max} = \frac{1}{2} m \frac{q^2 B^2 R_{max}^2}{m^2} = \frac{q^2 B^2 R_{max}^2}{2m} $

This energy is typically in the range of MeV (Mega-electron Volts).

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Limitations of the Cyclotron

The basic cyclotron has some limitations:

1. Relativistic Effects: As the particle's speed approaches the speed of light, its mass increases according to Einstein's theory of relativity ($m = m_0 / \sqrt{1 - v^2/c^2}$). This causes the time period $T = \frac{2\pi m}{|q|B}$ to increase with speed. The particle's revolution period no longer matches the fixed frequency of the alternating electric field (it falls out of resonance), and the acceleration stops. This limits the maximum energy achievable.
2. Maximum Radius: The maximum energy is limited by the physical size ($R_{max}$) of the dees.
3. Not Suitable for Electrons: Electrons have a very small mass. They reach relativistic speeds very quickly even at relatively low energies. Also, being lighter, they spiral much faster and radiate more energy when undergoing circular motion (synchrotron radiation). Therefore, cyclotrons are generally not used for accelerating electrons; linear accelerators are preferred.
4. Collisions: Collisions between the accelerated particles and residual gas molecules in the evacuated chamber can reduce the number of particles reaching full energy.

More advanced accelerators like synchrocyclotrons and synchrotrons overcome the relativistic mass increase problem by varying the magnetic field or the frequency of the accelerating electric field as the particle gains energy.

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