Electricity and Magnetism

Currents

Current is basically just the flow of charged particles. To have any type of flow of course we must consider time.

Current = Charge / Time (I=Q/t)

Current Density = Current / Cross sectional Area (J=I/A)

Electron Drift velocity = Current / (free electron per metre³ * Area * electron charge) (v=I/(nAq))

Power = current * potential difference = (potential difference)²/Resistance (P=IV and P=V²/R)

Ohms law states that: The current through a metallic conductor is directly proportional to the potential difference across it, provided the temperature and other physical conditions remain constant.
From this law we get: Voltage = (Current)*(Resistance) (V = IR)

Electrical Energy = Power * time (E = Pt)

Conductance = Current / Potential Difference (G=I/V)

Resistivity = Resistance * Cross Sectional Area / Length (ρ = RA/l)

Conductivity = 1 / Resisitivity (σ=1/ρ)

Circuits

This is a series circuit of three lamps

The current is the same at all points in the circuit

The total p.d (V) across the three lamps is equal to the sum of the individual p.ds across the lamps.

V = V₁+V₂+V₃

Dividing by I:
R_T = R₁+R₂+R₃

This is a parallel circuit of three lamps

The potential difference is the same across each lamp.

Since charge is conserved the total current is equal to the sum of the individual currents through each lamp.

V/R_T = V/R₁+V/R₂+V/R₃

Dividing by V:
1/R_T = 1/R₁+1/R₂+1/R₃

Kirchoff's Laws

Kirchoff's first law states the algebraric sum of the currents at a junction is zero.
Kirchoff's second law states in any closed loop of a circuit, the algebraric sum of the e.m.f.s is equal to the algebraric sum of the potential differences across all the resistences in the loop.

Electromotive force E.M.F: E = V + Ir

where r = internal resistance

Magnetic Fields

The Magnetic field from the centre of a wire: B = μ₀I/2πr

where μ₀ is permeability of free space = 4π*10^-7

The Magnetic field from a Long solenoid: B = μnI

where n = number of turns

The Magnetic field from a flat circular coil: B = μIn/2r

The Force on a current carrying wire: F = BIl

The Force on a charge moving in Magnetic field: F = Bqv

The Radius of the circular path: r = mv/Bq

The Hall Effect is defined as: V_H = BI/nzq

where z = length of conductor

EM Induction

Magnetic Flux is: Φ = BA

Faradays Law states that the magnitude of the induced e.m.f is proportional to the rate of change of flux cutting, or to the rate of change of flux linkage.

Fradays law can be expressed as: E = n(dΦ/dt)

Lenz's Law states that the direction of the induced e.m.f is such that it tends to oppsoe the flux change which causes it, and does oppose it if there is an induced current.

Lenz's Law can be expressed as: E = -n(dΦ/dt)

Energy stored in an inductor: W = 1/2LI²

where L is the self inductance

The electirc motor formula: VI = I²R+EI

where VI is the power supplied to the motor
I²R is the rate of energy dissapation as heat
EI is the rate at which the motor performs mechanical work

Electrostatics

The Coulomb force exerted on two point charges is: F = Q₁Q₂/4πε₀r²

where Q₁ and Q₂ are the point charges
ε₀ is the permitivity of free space = 8.85*10^-12 Farad m^-1

The Field strength at a point in an electric field is the force per unit charge exerted by the field at that point.

The Field Strength is: E = F/q

The Field Strength due to a point charge: E = (1/4πε₀)(Q/r²)

The Electric Potential at any point in an electric field is defined as the work done(W) per unit positive charge moved from infinity to the point.

The Electric Potenitial can be defined as: V = W/Q

The Electric Potential due to a point charge:V = (1/4πε)(Q/r)

The relationship between field strength and potential is: E = -V/d

where d is the distance of separation

Storing Charge

Capacitance C is defined as:C = Q/V

The Capacitance of a simple parallel-plane capacitor is:C = εA/d

The relative permittivity of the dialectric ε_r is: ε_r = ε/ε₀

For a parallel-plate capacitor with a dialectric of relative permitivity ε_r then:C =(ε_rε₀A)/d

The energy of a charged capacitor is:W = 1/2CV²

For Capacitors in parallel:C_T = C₁+C₂+C₃

For Capacitors in series:1/C_T = 1/C₁+1/C₂+1/C₃

The time constant of a CR circuit is:T = CR

Alternating Current

V = V₀sinωt and I = I₀sinωt

where V₀ and I₀ are the peak values of p.d and current

For an a.c waveform, the r.m.s or effective value of current(or voltage) is defined as the steady, direct value of the current(or voltage) which converts electrical energy to other forms at the same rate as the alternating current(or voltage).

For a sinusoidal alternating current:I_rms = I₀ /√(2) and V_rms = V₀ /√(2)

For a Transformer:V_s/V_p = N_s/N_p

where V_s is the secondary volts
V_p is primary volts
N_s is the secondary turns
N_p is the primary turns

AC circuits

Capacitive reactance is given by:X_c = 1 / (2πfC)

Inductive reactance is given by:X_L = 2πfL

For a R-C series circuit the impedance is given by:Z = √(R²+X_c²)

The current leads the applied p.d by a phase angle of:Φ = tan^-1(X_c/R)

For a R-L series circuit the impedance is given by:Z = √(R²+X_L²)

The current lags the applied p.d by a phase angle of:Φ = tan^-1(X_L/R)

For a R-L-C series circuit the impendance is given by:Z = √(R²+(X_L-X_c)²)

The current lags the applied p.d by a phase angle of:Φ = tan^-1((X_L-X_c)/R)

Electricity and Magnetism