CORE 7: FIELDS & THEIR CONSEQUENCES
7.1  FIELDS 
7.2  GRAVITATIONAL FIELDS 
7.3  ELECTRIC FIELDS 
7.4  CAPACITANCE 
7.5  MAGNETIC FIELDS 
7.6  ALTERNATING CURRENTS AND TRANSFORMERS 
1. FIELDS
# a force field is a region where a body experiences a noncontact force
# 3 types of force field exist: gravitational, electric, magnetic
# Newton’s Law: the force between 2 point masses is proportional to the product of
the masses and inversely proportional to the square of their separation.
F = Gm_{1}m_{2 }/ r^{2}
# G is called the gravitational constant, units N m^{2} kg^{ 2}
# Gravitational field strength g, at a point in a field, is the force that would act
on 1 kg placed at that point.
g = F / m , units are N / kg
# Field strength is a vector: get neutral points between two fields
# variation of g for a radial field (eg away from the centre of the Earth): g = GM / r^{2}
^{ }
# Equivalence of g as F/m and as acceleration of free fall.
# Gravitational potential, at a point in a field, is the work done in bringing
1 kg up from ¥ to that point.
V = W / Q , units are J / C.
# potential at any point in a radial field, V = − GM / r
# Gravitational potential energy = mass x gravitational potential.
Gpe = − mGM / r
# Work Done, W, in moving a mass in a field: W = m DV
# Equipotential surfaces. No work is done when moving along an equipotential surface.
# Graph showing how gravitational potential V varies with distance r away (in a radial field):
# Field strength = − (potential gradient), g = − DV / Dr so, field strength at any point
on the graph = the gradient at that point.
# DV = area under the g against r graph.
# Geostationary satellites : 24 hours, equatorial, about 42 000km orbit
radius (equivalent to an orbit height of 36 000km), west to east.
# Exploration satellites : orbits much closer in, shorter period eg few hours,
Can be polar orbit then all of globe can be scanned by single satellite.
# T^{2} ∝ r^{3} , applied to orbits (Keplers 3rd law)
# use of logarithmic plots to show relation between T and r
you need to be able to show how Newton's equations agree with Keplers 3rd law:
# kinetic energy, E_{k} = ½ m v^{2} = ½ m G M / r
total energy = potential energy + kinetic energy
ETotal = Ep + Ek = –G M m / r + ½ m G M / r = − ½ m G M / r
# Escape velocity (v_{e}) is the minimum velocity of projection for an object to escape the Earth’s gravitational field
using the KE to GPE principle, ½ m v2 = (–) G M m / R , v_{e} = (2 G M / R)^{1/2}
# Coulomb’s Law: the force between 2 point charges is proportional to the product
of the charges and inversely proportional to the square of their separation,
F = Q_{1} Q_{2} / 4πε_{0 }r^{2}
where ε_{0 } is called the permittivity of free space, having units F / m
# Electric field strength E, at a point in an electric field, is the force that would
act on 1C placed at that point.
E = F / Q units are N / C
# Variation of E away from the centre of a charged sphere (radial field):
E = Q / 4πε_{0 }r^{2 }

# E between parallel plates (uniform field):
E = V / d (other unit for E is V / m) 
# Electric potential, at a point in a field, is the work done in bringing
+1 C up from ¥ to that point.
V = W / Q , units J / C or Volts V
# Electric potential in a uniform field, V = Q / 4πε_{0 }r.
Graph showing how electric potential varies with distance away (in a radial field):
# Electric Potential Energy = charge x electric potential,
Epe = QV, Epe = qQ / 4πε_{0}r for a charge q in a radial field.
# Work Done, W, in moving a charge q in a field: W = qDV
E = V / d derived from work done, F x d = q DV
# Field strength = ±(potential gradient) , (NB electric fields can be attractive or repulsive)
E = ± DV / Dr units V / m
# Charged particles moving at right angles to an electric field follow a parabolic
trajectory, if the field is uniform the particles experience a constant force.
Equations for uniform acceleration apply here: SUVATS!
Charged particles moving parallel to the E field experience no force.
4. CAPACITANCE
# Capacitance of a capacitor = charge on 1 plate divided by p.d. between plates
# dielectric equation, C = A ε_{r } ε_{0 } / d
where ε_{r } is called the relative permittivity or dielectric constant
# when a dielectric is inserted it becomes polarised, reducing the resultant electric field between the plates
which in turn increases the capacitance of the capacitor
# time constant = R x C (units seconds)
# Charging capacitor: time constant = time taken for a capacitor to charge to 63%
of final (max) charge or voltage
# Discharging capacitor: time constant = time taken for a capacitor to discharge
to 37% of initial charge or voltage
# Note that the half charge time (or half discharge time) is 0.69 RC
# capacitor charge equation use, Q = Q_{0} ( 1 − e ^{  t / RC} ), capacitor discharge equation use, Q = Q_{0} e ^{  t / RC}
# Energy stored by a charged capacitor: E = ½ QV from area under the Q / V graph.
Also, E = ½ C V^{2} and E = ½ Q^{2} / C.
# Graph work: you need to know a whole range of graphs for both charging and discharging: V / t , I / t, Q / t, E / t
Also, watch out for the log graphs: ln Q / t , ln V / t
# Fleming’s left hand rule : thumb = force direction, 1^{st} finger = field direction, 2^{nd} finger = current direction:
F = B I L
# Magnetic flux density of a field = the force that acts on a wire of length 1 m
carrying a current of 1 A whilst lying at right angles to the field.
1 Tesla = 1 N A^{1} m^{1}
# A charged particle moving in a B field follows a circular trajectory because
a force continually acts at right angles to its direction – the condition for
circular motion. This is a centripetal force.
F = B Q v, B Q v = mv^{2} / r
# Magnetic flux F threading a surface is the flux density normal to that surface
x the area of that surface: F = B A
1 Weber = 1 Tesla x 1 m^{2}
# Magnetic flux linkage is the flux density normal to a coil x area of coil x
number of turns on coil: N F = B A N
# Faraday’s Laws (combined statement) the emf induced in a conductor cutting
through field lines = the rate of flux cutting. For a coil of N turns this is written:
E = DNF / Dt
# Lenz’s Law the direction of the induced emf/current is always so as to oppose
the effect that caused it. (this is a statement of the law of conservation of energy)
NB. In the diagram above, the coil becomes an induced electromagnet (Faraday)
whose upper pole is a North (Lenz).
Induced currents are often called eddy currents.
# AC generators: in a coil rotating at ω rad/s in a uniform B field:
In Figure1 above, the flux linking the coil = BAN cosθ ,
where θ = angle between the normal to the coil and the magnetic field.
also, θ = ωt , and when Faraday's E = DNF / Dt is applied we get:
# EMF induced in the coil, E = BANω sinωt
NB. In figure 2 flux linking the coil is a minimum but E is a maximum in this position.
6. ALTERNATING CURRENTS
# SINUSOIDAL VOLTAGE: one for which a graph of pd against time is a sine curve
# INSTANTANEOUS values vary continuously throughout 1 cycle.
PEAK (maximum) values are more useful
# FREQUENCY = 1 / T
# ROOT MEAN SQUARE VALUE (RMS) : The rms value of an alternating current is the value of the
DIRECT current which would produce the same power.
ADVANTAGE : dc theory may be used on rms values
# RMS AND PEAK values are related:
V_{RMS }= V_{0}_{ }I_{RMS }= I_{0 }
_{ Ö2 } _{ Ö2}
# Oscilloscopes can be used to measure peak values directly from which rms values can be calculated,
using the equation above.
Peak voltage = height of trace x voltage gain(in V/div)
Oscilloscopes can also be used to measure frequencies,
by first measuring the TIME PERIOD T of the signal, then using f = 1 / T.
Time period = horizontal length of 1cycle x timebase setting
# TRANSFORMERS: turns ratio equation , V_{s } / V_{p }= N_{s} / N_{p} ,
for an ideal transformer, Power out = Power in, I_{s} V_{s} = I_{p} V_{p}
in practice, Power out < Power in , so I_{s} V_{s} = ε I_{p} V_{p}
where ε = efficiency of the transformer (ideal transformer ε = 1)
# Be familiar with the causes of transformer inefficiency.
# Be familiar with causes of power losses in transmission lines.