10.1 Describing Fields

A space shuttle of mass 2 × 10⁶ kg is travelling from the Earth to the moon. It accelerates uniformly from launch at 5.25 m s^–2.It has enough propellant to provide thrust for the first 124 seconds.

The mass of the Earth is 5.97 × 10²⁴ kg and the mean radius is 6.37 × 10⁶ m.

Calculate the work done by the rocket during the first 124 seconds after launch. State any assumptions you have made.

The Moon has a radius approximately 27% that of the Earth, and a mass of 1.2% that of the Earth. 

Calculate the gravitational potential at the surface of the moon in terms of the gravitational potential on the surface of the Earth.

The gravitational field strength lines between the Earth and the moon can be drawn on a diagram.

Point P is the neutral point between the Earth and the moon where there is no resultant gravitational field.

Sketch the equipotential lines between the Earth and the moon on the diagram.

Imagine that it was possible to construct a tunnel through the centre of the earth, connecting a point on the surface to the diametrically opposite point. Assume that the earth is perfectly spherical with an evenly distributed mass and that the mass and volume of the tunnel, air resistance and friction are negligible.

Describe the variation in gravitational field strength and how speed of travel would vary if a person were to jump into the hole.  

Within a hollow sphere of uniform density, the gravitational field strength is zero. 

Using this information, derive an expression for the gravitational field strength at any point in the tunnel in terms of:

• The distance from the centre of the earth = r
• The radius of the earth = R_e
• The gravitational field strength on the surface of the earth = g_surf

This case is analogous to a mass bouncing on a spring where , and where in this situation .

The time period for a mass on a spring, T, is equal to . For a satellite in orbit at a distance r, from the centre of a large body with mass M, the orbital speed can be obtained as .

Using this information, show that the time for a traveller to reach the other end of the tunnel is the same as the time taken for a satellite in orbit just above the surface of the Earth to travel through half its orbit.

Three charges are fixed at the corners of a right-angled triangle.

The length of both the horizontal and vertical sides is d.

Show that the electric potential at point P, halfway between the −2Q and −6Q charge is given by .

Before the discovery of quarks, scientists speculated that the subatomic particles might be made up of smaller particles.  
If an electron was made up of three smaller, identical particles with charge q, which are brought in from an infinite distance to the vertices of an equilateral triangle, it would have this arrangement.

The radius of an electron is 2.82 fm. 

Show that the work done in forming an electron consisting of 3 identical particles in this arrangement is given by:

In an electron gun, electrons are released from a cathode and accelerated towards an anode. The electrons leave the electron gun at 10% of the speed of light. 

Calculate the potential difference between the cathode and the anode. 

A science fiction film director is planning for a battle scene between two spacecrafts. The first spacecraft uses an electron gun to fire a beam of electrons at the second spacecraft from close range. The electron beam is created by accelerating electrons from rest between electrodes with a potential difference of 120 V.  

To shield against the attack, the second aircraft creates a uniform electric field around itself and the electrons are stopped after 85 m. The director wishes to calculate the strength of the electric field, but is not aware of the equation .

Calculate the strength of the electric field without the use of this equation.

After the failure of the first spacecraft to break through the electric shield of the second spacecraft a new weapon is to be designed. Instead of firing electrons, research is carried out to see if firing negatively charged ions with a charge of –2e and a mass of 2.26 × 10^-26 kg would be more effective. The second spacecraft uses the same electric field as in part (a) to shield itself and that the electric field is uniform.

Calculate the magnitude of the minimum velocity at which these ions would need to be fired if they are to strike the second spacecraft from a distance of 1 km.

Another option to attack the second spacecraft is to create a superweapon which can be fired from the aliens' home planet. As this weapon will fired from such a long way away from the second spacecraft, the spaceship and shield can be modelled as a charged particle carrying a charge of −130 C. The electrons fired by the superweapon have a kinetic energy of 12 MeV.

Calculate how close the electrons will come to the second spacecraft before they are stopped by the shield.

Two charged objects, A and B, are brought close together. The equipotential lines around each object is shown. 

[1]

Identify the electric charge on objects A and B and explain your answer. 

A different pair of equally charged objects, C and D, are brought close together. Equipotential lines at –10 V, –6 V and –2 V around C and D are shown.

Draw electric field lines between C and D. 

Objects C and D are now given a charge of –3.5 nC and –4.0 nC respectively. 

X is a position at –6 V which is equidistant from both C and D. 

Determine the potential due to object C at the position labelled X. 

Hydrogen atoms in an ultraviolet (UV) lamp make transitions from the first excited state to the ground state. Photons are emitted and are incident on a metallic, photoelectric surface as shown.

The photoelectric surface is grounded, and the variable power supply is adjusted so that the electric potential of the collecting surface is 1.5 V. 

Describe the properties of the electric field between the photoelectric surface and the collecting plate. 

On the diagram, draw and label equipotential lines at 0.5 V and at 1.0 V.

Electrons are released from the photoelectric surface and move toward the collecting plate. 

Determine the work done by the electric field on the electrons as they arrive at the midpoint between the photoelectric surface and the collecting plate. 

Describe the motion of the electrons between the photoelectric plate and the collector plate. Your answer should consider the field at the edges of the plates. 

A β^– particle is placed above a grounded metal plate. 

Sketch the electric field lines between the β^– particle and the metal plate. 

The β^– particle is replaced with an α particle at the same position above the grounded metal surface. 

Outline the similarities and differences between the electric field now created to that in part (a).

The grounded metal surface is removed in order to analyse the combined electric field created between the α particle and the β^– particle.

Sketch the electric field produced between an α particle and a β^– particle.

Discuss whether there is a point of zero electric field for the diagram in part (c).

The diagram shows the gravitational field produced by planet P.

Outline how this diagram shows that the gravitational field strength of planet P decreases with distance from the surface.

The diagram shows part of the surface of planet P. The gravitational potential at the surface of planet P is –6 V and the gravitational potential at point X is –2 V. 

On the grid, sketch and label the equipotential surface corresponding to a gravitational potential of –4 V. 

A meteorite, very far from planet P, begins to fall to the surface with a negligibly small initial speed. The mass of planet P is 3.0 × 10²¹ kg and its radius is 2.3 × 10⁶ m.

Calculate the speed at which the meteorite will hit the surface, assuming planet P has no atmosphere.

Without detailed calculation, state and explain the effect on the meteorite's impact velocity if planet P's mass was twice as large.