Special Relativity and Electric Fields

What is the electron volt?

The electron volt is a unit of energy defined to be the energy involved in moving an electron across a potential difference of 1V.

What is electric shielding?

Electric field lines do not exist inside any solid or hollow conductor, whether charged or uncharged, provided there is no charge inside the cavity.

What is an electric field?

An electric field is a region of space where a charged body experiences a force because of its charge.

What direction is acceleration in an electric field?

The acceleration is in the direction of the field for positive charges and against the field for negative charges.

What is work done equal to?

By conservation of energy, the work done on the charge is equal to its change in electric potential energy (EPE).

Why is the speed of protons constant inside dees?

No work is done on the protons as the dees are hollow conductors with zero internal electric field.

What is voltage (potential difference)?

The voltage between 2 points in an electric field is defined to be the work done per unit charge in moving a positive test charge between the two points.

What is electric field strength?

The electric field strength at a point is the electric force per unit charge experienced by a positive test charge placed at that point.

What is the strength of the electric field between two parallel plates dependent on?

The potential difference between the plates as well as the distance between the plates.

Describe and explain the motion of oil droplets that were not ionised by X-rays:

Oil droplets that were not ionised remained electrically-neutral and did not experience an electrostatic force in the field between the plates. Therefore, neutral oil droplets fell accelerated under gravity from the upper to lower plate.

Describe and explain how protons are accelerated in a cyclotron:

The electric field does work on the protons each time they cross the gap between the dees. The protons are steered repeatedly towards the electric field to increase their speed and kinetic energy.

What is a cyclotron?

A cyclotron is a particle accelerator in which ions are made to repeatedly cross an electric field, allowing them to be accelerated to high speeds.

What is the motion of a charged particle through uniform electric fields?

The motion of a charged particle in a uniform electric field is analogous to projectile motion. The difference is in the acceleration. While a_g = 9.8ms^-2 downwards for all objects, the magnitude of acceleration in an electric field depends on the strength of the field and also on the mass and charge of the particle.

What is an inertial reference frame?

An inertial reference frame is a frame that is moving at a constant velocity. This occurs when the frame has no net force acting on it and therefore Newton’s 1st law of motion applies.

What is a reference frame?

‘All motion is relative’

A reference frame is a perspective used to determine positions and velocities in that frame. Different frames move relative to one another, meaning observers in different reference frames will measure different velocities for the same object.

What are the two postulates of Einstein’s theory of special relativity?

1. The laws of physics are the same in all inertial reference frames.

2. The speed of light is an absolute constant; it is the same in all inertial reference frames.

What are muons?

Muons are subatomic particles which are produced in the atmosphere at altitudes of approximately 15000m with speeds of approximately 99.97% of the speed of light. They are unstable and have an average lifetime of 2.2ms.

What is time dilation?

For both postulates in Einstein’s theory of special relativity to be true, time must not be absolute and independent, but run at different rates in different reference frames, This is time dilation, with time passing more slowly in a reference frame which is moving relative to an observer.

What is length contraction?

Objects moving at relativistic speeds are shorter to observers in a ‘stationary’ reference frame. Similarly, the distance between two stationary points will be shorter when measured by an observer in a moving reference frame.

When there are two different ions travelling towards a detector, which one reaches the detector first?

The ion with the lower mass and greater acceleration in the electric field. This is because the time taken to reach the detector is inversely proportional to the acceleration of the ions, meaning the ion with the lower mass reaches the detector first.

Describe how the electric field near sharp points may ionise the air:

The electric field is strong at the sharp points of the aeroplane which causes the ionisation of atoms and molecules in the air. Ionised particles are then accelerated away from the conductor creating further ionisations in the air.

Why does lightning usually strike the aeroplane’s wingtips?

Charged particles in the air are more attracted to the wingtips of the aeroplane as the electric field is strongest near sharp ends of a conductor.

Why are passengers inside an aeroplane’s cabin unharmed when lightning strikes?

The outer casting of the plane is a hollow electrical conductor, therefore all charge is on the surface of the plane, and no charge enters the hollow cavity where it would cause harm to passengers.

State a simple method of increasing electron speed in a gyrotron:

Increasing potential difference increases electron soeed in a gyrotron.

What is the principle of superposition?

When 2 or more charged particles are present, the force on any one of the particles is the vector sum of the forces due to each of the other charges.

Why can’t objects with mass ever reach the speed of light?

As the object accelerates, it gains mass, and thus takes more and more energy to increase its speed. It would take an infinite amount of energy to make an object with mass reach the speed of light.

What is relativistic momentum?

As the speed of an object approaches the speed of light, the objects mass, and therefore momentum, increases exponentially.

What happens to the KE of ions in the dees?

While in the dees, the KE of the ions is constant, as there is no electric field within the dees as they are hollow conductors.

How do cyclotrons work?

Upon entering the dees, a magnetic field causes them to follow a circular path. The potential difference of the dees alternate, causing the direction of the electric field to reverse. When returning to the gap, the ions (now travelling in an opposite direction) are once again accelerated by the field, gaining another quanta of energy. This is repeated over and over, allowing ions to reach very high speeds.

Explain how it is possible for negative kaons to travel between two detectors:

The half-life of negative kaons is dilated in the reference frame of a stationary reference frame such as the stationary detectors. The extended half-life of negative kaons allows them to travel a greater distance when measured by an observer at rest in the stationary reference frame.

State and explain how the length of the linear accelerator measured in the reference frame of the electrons changes as their relativistic speed increases:

The length of the linear accelerator decreased in the reference frame of the electrons. The Lorentz factor increases with particle speed which results in a proportional decrease in the length measured in the reference frame of the electrons as length is inversely proportional to the Lorentz factor.

State why protons cannot be accelerated to the speed of light:

The momentum of the protons approaches infinity as its speed approaches c. To change the momentum of the protons to infinity requires infinite force and work, which is impossible.

Describe the effects on the momentum of a proton when it moves at relativistic speeds:

Momentum increases with the relativistic speed of the protons. Therefore, eventually, the relativistic momentum approaches infinity when v approaches c.

Describe and explain the path followed by a proton in the uniform electric field:

The proton experiences a vertical force but no horizontal force in the electric field. The horizontal component of velocity is constant, and the vertical component is changing, which results in the proton following a parabolic path.

Why is the motion of electrons similar to a projectile in the absence of air resistance?

The electron and projectile experience a vertical force due to their respective fields but no horizontal force. This causes the electron and projectile to follow a parabolic path in their fields.

State the advantage of having a hollow conductor surrounding the inner wire of a cable transmitting electrical signals from an antenna to a TV:

An excess charge would move onto the inner conducting wire and alter the magnitude of the current in each pulse, resulting in poor TV reception.

Explain the evidence that the component of the electric field parallel to the surface of the hollow conductor is zero:

The charges on the surface of the conductor are in electrostatic equilibrium, and each charge is stationary. If a component of the field were horizontal to the surface of the conductor, the charges would accelerate and move in that direction.

State the magnitude of the electric field produced by the hollow conductor when the surface charge increases:

The magnitude of the electric field increases because the electric field strength is directly proportional to the magnitude of the field producing charge.

What is corona discharge?

Electric fields near sharp points may be strong enough to ionise molecules in the air, resulting in a flow of charge away from the conductor.

Why does a cyclotron use an alternating potential difference between the dees?

An alternating potential difference ensures the proton moves across the gap between the dees in the direction of the electrostatic force.

How does corona discharge occur?

When a strong electric field polarises nearby air molecules which are then attracted to the conductor. Upon contact, charge transfers and the molecule is ionised. This causes the molecule to have the same charge as the conductor, and it is thus repelled. The net effect is charge movement away from the conductor.

What are ‘sharp points?’

When electric field lines concentrate at points of maximum curvature.