Week 7 and 8

Basic Facts

When electric charges move they create an electric Current. When the charges flow in wires they are best described by the amount of charge passing a point on the wire in each second, the current I.

I = number of coulombs of charge passing a point in 1 second (units: 1 Ampere = 1 Coulomb/sec)

Ohms Law: Experimentally, we find that when a current flows in a conductor it creates a potential difference V between the ends of the conductor and the potential drop is proportional to the current

V = I•R where the constant is called the Resistance of the conductor. It is measured in Ohms where
1 Ohm = 1 Volt/1 Ampere.

When two conductors of resistances R1 and R2 are joined in series then they form a single resistor of value Rt = R1 + R2.

When a current flows in a long straight wire it generates a Magnetic Field around it. The field lines form concentric circles round the wire and the strength of the field decreases with distance.

Superposition: the magnetic field at any point in empty space is given by the sum of the magnetic fields from various sources.
NOTE: this is NOT true in materials!

Magnetic Dipole: A current I flowing round a loop of area A, the direction being chosen so that the current flows around the perimeter of the loop according to a right-handed rule, form a Magnetic Dipole with dipole moment m = IA.

The magnetic field strength a distance r from the axis of very long straight wire carrying current I is given by

so long as r is outside the wire.

Lorentz Force Law: a particle of charge q travelling at velocity v in a region of space where there is an electric field E and a magnetic field B feels a force F_L given by

As a special case, the force on a straight length of wire s carrying a current I through a region where there is a uniform magnetic field B is given by

(Note that s is the vector length of the wire, I didn't use lower case L as books usually do because it ends up looking ike a numeral.)

Useful Facts

If the charge is distributed in space then we describe it by a current density, J. The current density is the amount of charge crossing unit area in 1 second

J = I/area (units: Amperes/m²)

Ohm's Law: When an electric field is applied to a conducting material a current flows. In most materials, at least at low currents and fields, the current density is proportional to the field

J = σE where the constant of proportionality, σ is called the Conductivity and is a property of the material used. We sometimes write this in reverse

E = ρJ where the constant ρ, called the Resistivity, is given by ρ = 1/s.

The Law of Biot and Savart: the magnetic field caused by a wire that carries a current I along a closed path L is given by
where dl' is an elementary vector pointing along the path at the point r' and r is the vector pointing to the point at which the measurement is being made.

The force on a general piece of wire following a path Γ through a non-uniform B field is given by
F = ∫I dl×B where the integral is along the path Γ.