CORE 7: FIELDS & THEIR CONSEQUENCES

CONTENTS

 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 non-contact 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 = Gm1m2 / r2

# G is called the gravitational constant, units N m2 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 / r2

# 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.

#  T2   ∝  r3 ,  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,  Ek  =   ½ m v2   =   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 (ve) 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  ,   ve  =   (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 = Q1 Q2 / 4πε0 r2

where  ε0  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 / 4πε0 r2 # 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 / 4πε0 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 / 4πε0r  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.

# Capacitance of a capacitor = charge on 1 plate divided by p.d. between plates

# dielectric equation,  C  =  A εr ε0  / d

where εr 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  =  Q0 ( 1  −  e  - t / RC ),   capacitor discharge equation use,  Q  =  Q0  e  - t / RC

# Energy stored by a charged capacitor:  E = ½ QV from area under the Q / V graph.

Also,  E = ½ C V2  and  E = ½ Q2 / 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, 1st finger = field direction, 2nd 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  =  mv2 / 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 m2

# 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 / D

# 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

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:

VRMS = V0                           IRMS = I0

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 /  Vp   =   Ns / Np  ,

for an ideal transformer,    Power out = Power in,     Is Vs  =   Ip Vp

in practice,        Power out < Power in , so    Is Vs  =  ε Ip Vp

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.