2Nd Derivation:

Mean lifetime

The mean lifetime t_mean, generally called the lifetime, of nucleus or unstable particle is proportional to the half-life t_1/2:

In particle physics the life of an unstable particle is usually described by the lifetime, not the half-life.

Avogadro’s number

There are atoms in 1 mole.

Mass = (number of moles) × (molar mass)

1 mole of has a total mass of A grams.

Natural Radioactivity

• Series starts with ²³²Th

• Processes through a series of alpha and beta decays

• The series branches at ²¹²Bi

• Ends with a stable isotope of lead, ²⁰⁸Pb

Radiation Detectors

• Mica window

• Low pressure gas

• High voltage

• Anode/cathode

• High E field

• Massive ionization

• Electron avalanche

• Pulse

• It detects individual radiation particles or photons;

• It can be connected directly to a counter;

• Radiation enters the tube through the thin mica window (good for α – particles), and ionizes the low pressure gas, e.g. air or argon;

• Electricity conduction is improved at low pressure;

• The free electrons are accelerated towards the anode by the high voltage producing more ionizations and free electrons;

• An avalanche of electrons arrives at the anode, which is detected as a voltage pulse that can be counted electronically.

Mass defect and Binding energy

ΔM (mass defect) =M_REACTANTS – M_PRODUCTS

Work must be done to remove any nucleon from a nucleus.

The nuclear binding energy ( ) is the energy equivalent of the mass defect. It is the energy holding the nucleons together. If is added to the nucleus, the nucleus will split into its parts

Δ E = Δ M c²

Note: Units of “ ” are usually given in MeV

More convenient to use ‘binding energy per nucleon’.

Binding Energy per nucleon =

Nuclear Reactions including Reaction Energy

A nucleus X can be bombarded by a particle a, resulting in a daughter nucleus Y and an outgoing particle b:

The reaction energy Q is defined as the total change in mass-energy resulting from the reaction

• If > 0, reaction is exothermic and energy is expelled to surroundings;

• If < 0, reaction is endothermic and energy is absorbed from surroundings.

Energy Difference of Fusion and Fission

• Effect of fusion and fission

• Both decrease the average nucleon PE (increasing binding energy per nucleon). PE lost is emitted as gamma rays and KE of particles.

Spontaneous and induced fission

Spontaneous – occurs naturally

Induced – requires a ‘slow’ neutron to react with nucleus

Nuclear reactors – controlled fission

• Nuclear reactor is a system in which controlled nuclear chain reaction is used to liberate energy.

• Fuel rods - Long tube containing pellets of fissionable material, which provide fuel for nuclear reactors.

• Moderator – slows down the high-energy neutrons.

• Control rods – control the rate of the reaction (absorb neutrons without any additional reaction).

• Coolant - removes heat from the reactor core and transfers it to electrical generators and the environment.

Fusion

Important as energy source in the core of stars.

Typically there is conversion of Hydrogen nuclei to heavier (He, N, O, C, etc) through fusion (this takes place at very high temperatures , followed by immediate release of energy.

Typical Fusion Reactions

Proton – proton (hydrogen – hydrogen) reactions

Fusion has been used uncontrolled in Hydrogen Bombs. Often a fission detonator is used to create high temperatures and start fusion process.

A challenge today is to create a controlled fusion reaction. The main problem is the high temps required to initiate fusion (this is called confinement) One interesting reaction is:

Forces due to gravitation and charge

Newton’s law of Gravitation Two particles m1, m2 distance r apart	Coulomb’s law Two charges q1, q2 (in a vacuum) distance r apart

Constants

G is the Universal Gravitational constant

G = 6.67 × 10^-11

ε₀ is the permittivity of free space

ε₀ = 8.85 × 10^-12

k = 8.99 × 10⁹

Field

• Field is a region in which a body experiences a force as a result of the presence of some other body.

Gravitational Field

Electric Field

Electric Field due to point charges

For fields due to more than one charge, we use the principle of superposition. That is, at any given point in space, we add all E-fields as vectors.

• The flux density and therefore the E field obey inverse square laws;

• Field lines start on positive charges and end on negative charges.

• Field lines can never cross.

Field lines, flux and flux density

• Electric Field lines or “lines of force”.

• The tangent to a field line is the direction of the E field at that point.

• The flux through an area is the total number of field lines that pass through the area.

• The field lines are closer together where the field is greater, i.e. the flux density is greater.

• Flux density is the number of lines of force per unit area where the area is perpendicular to the lines.

Gravitational Field

The gravitational field, g, due to a point mass

Kepler’s laws

1. The orbit of every planet is an ellipse with the Sun at one of the two foci.

2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3. T he square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. (Prove for a circular orbit)

Electric Potential Energy

The change in electrical potential energy of a system of two charges q₁ and q₂, as q₂ is brought from an infinite distance to a distance r relative to q₁ :

Prove that:

Electric Potential

Electric potential V_E(r) is U_E(r) per unit charge.

Derivation 1: Potential due to a point charge Q

Electric potential difference between any two points

Potential difference is work done against the electric field to move unit charge from one position to another position.

Consider a constant E field region: e.g., between the plates of a capacitor.

E is the negative of the gradient of V, Thus, the scalar field V can be used to find the vector field E

E quipotential lines due to point charges

Gravitational Potential Energy

The gravitational potential energy of a system of two masses m and M, as m is brought from an infinite distance to a distance r:

• Potential energy of mass m near the Earth

Change of potential energy in moving upwards to height h.

Gravitational potential

• Gravitational potential V_G(x) is the gravitational energy per unit mass.

Derivation 2: Gravitational potential V_g due to sphere of mass M (e.g., Earth)

Recall and derive

Electric Current

• Electric current is the rate of flow of charge through some region of space

• The SI unit of current is the ampere (A) 1 A = 1 C / s

• The symbol for electric current is I

Average Electric Current

• Assume charges are moving perpendicular to a surface of area A

• If +Q is the amount of charge that passes through A in time t, then the average current is: