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

 

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  2. GRAVITATIONAL FIELDS

       # 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

 

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  3. ELECTRIC FIELDS

       # 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.

 

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  4. CAPACITANCE

 

       # 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

 

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  5.  MAGNETIC FIELDS

 

      # 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.

         

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