Outline  Potential Energy, Potential,
and Capacitors


 Energy
 Potential Energy Difference, U_{B}  U_{A},
between points A and B equals the work done W_{AB}
carrying a positive test charge +q_{o} from A to B
without increasing its kinetic energy.
U_{B} – U_{A } = W_{AB}
= _{A}∫^{B }F ^{.} ds,
where F is the force you apply.
For no change in kinetic energy, the magnitude of the object's
velocity must remain constant so that there is no acceleration.
For no acceleration, the net force acting on the object is
zero.
F_{net} = F + F_{e}
= F + q_{o}E = 0 or
F =  q_{o}E and
U_{B}  U_{A } = W_{AB}
=  q_{o} _{A}∫^{B }E
^{.} ds.
 Potential Difference, V_{B} – V_{A}
= V_{BA }= (U_{B} – U_{A})/q_{o
}= (q_{o} _{A}∫^{B }E
^{.}ds)/q_{o}
V_{B}  V_{A} =  _{A}∫^{B
}E ^{.} ds.
 Potential Energy U at a point P equals the potential energy
difference between point P and some point where the potential
energy is taken to be zero.
 Potential V at a point equals the potential energy U per
unit charge.
V = U/q_{o}.
 Special Cases
 Special Cases for Potential Energy Difference and Potential
Difference
 Point charge q in field of a point charge +Q
 Potential Energy Difference U_{B}  U_{A
} = kqQ(1/r_{B} – 1/r_{A}),
where r_{B} is the distance of Q from point
B
and r_{A} is the distance of Q from point
A.
 Potential Difference V_{BA }= (U_{B}
 U_{A})/q = kQ(1/r_{B} – 1/r_{A})
 Point charge q in field of a parallel plate capacitor
with +Q and  Q charge over area A and plate separation
d.
 U_{B}  U_{A} = q(E)d = q(Q/A ε_{o})d,
where 1/4πε_{o}
= k = 9.0 x 10^{9} Nm^{2}/C^{2}.
 V_{BA }= (U_{B}  U_{A})/q
= Qd/A ε_{o
}
 Special Cases for Potential Energy and Potential
 In field of point charge take U at infinity equal to
zero. Then at point P a distance r from +Q the potential
energy U = kqQ/r. If the charge is negative, U =
 kqQ/r.
 Since Potential V = U/q, for +Q, V
= kQ/r and for  Q, V =  kQ/r.
 Since potential energy and potential are scalar
quantities, we can use the negative sign to indicate
a potential energy or potential less than zero. For
example, if you move a +q away from the  Q that
sets up the field without changing its kinetic energy,
you do work. Thus the potential energy at infinity
must be greater than at the point closer to the negative
charge. This can only be true if the potential energy
is negative near the negative charge  Q.
 If you have more than one charge setting up the field,
you find the potential at a point due to each one and
then add the potentials algebraically.
 Sample problems in 106
Problem Set for Potential Energy, Potential, and Capacitors:
14.
 Use of Conservation of Energy in solving problems
When energy is conserved,
U_{B } + K_{B} = U_{A } +
K_{A} or
U_{B }  U_{A} = K_{A }  K_{B}
Since by definition, V_{BA }= (U_{B } 
U_{A})/q, qV_{BA} = K_{A }  K_{B}
Sample problems in 106
Problem Set for Potential Energy, Potential, and Capacitors:
512.
 Capacitors
 Definition C = q/V_{BA
}
 For a parallel plate capacitor, V_{BA} = qd/A ε_{o}
and C = q/V_{BA} = A ε_{o}/d.
Notice that C is a constant that depends only on the dimensions
of the parallel plates and the constant ε_{o}.
 A charged sphere acts as though all the charge were
at the center of the sphere. At the surface of a
sphere of radius R,
V = kq/R and C = q/V = R/k.
 When a dielectric of dielectric constant κ is
inserted into a capacitor, its capacitance goes up by
a factor of κ.
 The potential energy stored by a capacitor U = 1/2 qV_{BA}
= 1/2 C (V_{BA})_{ }^{2} = 1/2 q^{2}/C.
 Note on units: 1 F = 1 C/V. For U = 1/2 C (V_{BA})_{
}^{2}, units of U are (C/V)(V)^{2} =
C  V = J. For U = 1/2 q^{2}/C, units of
U are C^{2}/(C/V) = CV = J. Sorry that coulomb C
and capacitor C are confusing.
 Capacitors in series and in parallel
 For the three capacitors wired in series (Fig. 1a above),
 The charge q is the same on each.
 The potential difference across each is not the
same unless the capacitance of each is the same. From
conservation of energy,
V_{ab} = V_{ac} + V_{cd}
+ V_{db
} q/C_{equ} = q/C_{1} + q/C_{2}
+ q/C_{3
}
 1/C_{equ} = 1/C_{1} + 1/C_{2}
+1/C_{3}, where C_{equ} is the equivalent
capacitor which replaces the three individual capacitors
and has the same charge q and the same potential difference
V_{ab}. For capacitors in series, the reciprocal
of the capacitances add to equal the reciprocal of
the equivalent capacitance.
 For the three capacitors wired in parallel (Fig. 1b
above),
 The potential difference across each is the same:
V_{a'b'}= V_{ab} = V_{a"b"}
 From conservation of charge, q = q_{1}
+ q_{2} + q_{3
}
 Since q = CV_{ab},
C_{equ}V_{ab} = C_{1}V_{ab}
+ C_{2}V_{ab} + C_{3}V_{ab}
or
C_{equ} = C_{1} + C_{2}
+ C_{3}.
For capacitors in parallel, the sum of the individual
capacitances equals the equivalent capacitance.
 Sample problems in 106
Problem Set for Potential Energy, Potential, and Capacitors:
15, 1720.

