Electric and Gravitational Potential Energy for a Uniform Field

1
Chapter 16
2
Electric and Gravitational Potential Energy for a
Uniform Field

• Comparison of Electric and Gravitational Field

3
Electric and Gravitational Potential Energy for a
Uniform Field (cont.)

• Comparison of Electric and Gravitational Field
  (continued)

4
Electric and Gravitational Potential Energy for a
Uniform Field (cont.)

• Work done by both Gravity and Electric Field
• The amount of work that must be performed by an
  external force to move a charge Q from B to A, a
  distance d in a uniform electric field E is
• W Q E d from W F x d
• On the other hand, work done by the Electric
• Field Force is
• Welectric - Q E d

5
Electric and Gravitational Potential Energy for a
Uniform Field (cont.)

• Types of Work
• Work ? F and ? are in the same direction
• Work ? F and v are in opposite directions
• Zero Work ? i) F, d 0
• ii) F and v at 90 angle

6
Electric Potential Energy for a Uniform Field

• Change in Electric Potential Energy
• The change in Electric Potential Energy from B to
  A
• ? EPE Q E d

7
Electric Potential Difference

• Electric Potential Difference between A and B
• The electric potential difference between points
  A and B, Va Vb is defined as the change in EPE
  of a charge q moved from A to B, divided by the
  charge Q (that is, change in EPE per unit
  charge).
• ?V Vb Va ?EPE / Q E d
• Type of Quantity Scalar
• SI-Units J/C 1 volt

8
Electric Potential Difference (cont.)

• Concept Electric Potential Difference represents
  the energy required to move a unit charge (1
  Coul) from one potential to another.
• Units Volt (V)
• ?V energy per unit charge
• Since ?V EelectricPE / Q
• Then EelectricPE ?V Q
• And EelectricPE Q E d

9
Electric Field

• Electric Field
• E ?V / d
• Units N/Coul or V/m

10
Electric Potential due to a Point Charge

• Electric Potential due to a point charge
• If the point of zero potential is taken to be at
  an infinite distance from a charge, then by
  calculus, the electric potential due to a point
  charge q at any distance r from the charge Q is
  given by
• Vp k Q/r

11
Electric Potentialdue to a Point Charge (cont.)

• Consider the sign of charge
• If Qp Q then Vp k Q/r
• If Qp -Q then Vp - k Q/r

12
Superposition Principle

• Superposition Principle
• The Electric Potential of 2 or more charges is
  obtained by this principle which states that the
  total electric potential at some point P due to
  several point charges is the algebraic sum of the
  potentials due to the individual charges

13
Electric Potential due to a System of Point
Charges

• Potential Energy of a pair of point charges

If V1 is the electric potential due to charge Q1
at point P, then the work required to bring a
charge Q2 from infinity to the point P is Q2V1.
This is the same as the Potential Energy of the 2
particle system separated at a distance r12. U12
PE Q2 V1 k Q1 Q2 / r12 Note Maintain sign
of each charge!
14
Potentials of Charged Conductors

• Potentials of charged conductors
• For a conservative uniform force field we
  established that
• W - PE
• PE Q (Vb Va)
• Therefore W - Q (Vb Va)
• If Va Vb, we obtain W 0
• This result tells us that no work is required to
  move a charge between 2 points that are at the
  same potential.

15
Potentials of Charged Conductors (cont.)
16
Potentials of Charged Conductors (cont.)

• To show that the Electric Potential of a charged
  conductor in electrostatic equilibrium is
  constant everywhere inside the conductor
• We established that W Q E d
• Inside a conductor E 0 therefore W 0
• EPE q (Vb Va) 0 therefore Vb Va
  constant (1)

17
Potentials of Charged Conductors (cont.)

• 2. To show the Electric Potential at the surface
  of a charged conductor is constant
• Consider the points C and D on the surface of
  the charged conductor. Since the displacement
  path CD is perpendicular to the Electric Field at
  the surface
• W 0
• W Q (VD VC) so if W 0
• then VD VC constant (2)

18
Potentials of Charged Conductors (cont.)

• Combining (1) and (2)
• The Electric Potential is constant everywhere
  inside a conductor and equal to its value at the
  surface.

19
The Electron Volt

• The Electron Volt
• Is the energy that an electron gains when
  accelerated through a potential difference of 1
  Volt (1J/C).
• Charge on an electron is 1.6 x 10-19
  Coulombs.
• ?V 1 Volt 1 J/Coul and Q 1 e
• EPE Q ?V ? EPE 1 eV
• So, 1 eV 1.6 x 10-19 C. x 1 Volt
• 1.6 x 10-19 C. x 1 J/C
• 1.6 x 10-19 Joules
• energy an electron gains when going
  through a potential difference of 1 Volt

20
EquiPotential Surfaces

• EquiPotential Surface
• A surface in which all points on it are at the
  same potential.
• Since all points are at the same potential then,
• ?V Vb Va 0
• but W Q ?V
• ? hence no work is done in moving a charge at
  constant speed on an equipotential surface.

21
EquiPotential Surfaces (cont.)

• Relationship between an EquiPotential Surface
  with E
• An EquiPotential Surface associated with any
  charge distribution is always perpendicular to
  the electric field at any point.

22
Capacitors

• Capacitor Composition
• A capacitor consists of 2 conductors whose
  charges are equal in magnitude but opposite in
  sign and separated by a given distance.
• Examples i) parallel plate capacitor
• ii) cylindrical capacitor
• iii) spherical capacitor

23
Capacitance

• Capacitance
• The capacitance C of any capacitor is defined
  to be the ratio of the magnitude of the charge Q
  on either side of either conductor to the
  potential difference V between them.
• C Q / ?V Q a ?V
• C constant of proportionality Capacitance
• Type of Quantity Scalar
• SI-Units Coul/V 1 Farad

24
Capacitors (cont.)

• Uses of a Capacitors
• To tune the frequency of radio receivers
• Short-term energy storing devices
• Store charge for later use in a camera or as in
  energy back up in computers if power fails
• Capacitor blocks surge of charge or energy to
  protect electric circuits
• Defibrillator
• Surge Protectors (UPS)

25
Parallel-Plate Capacitor

• Parallel-Plate Capacitor
• From the geometry of the conductor, Capacitance
  of a parallel-plate capacitor is given by
• C Capacitance
• A area of plate
• d distance of separation
• ?o permitivity of free space s (vacuum)
• 8.85 x 10-12 Coul2/Nm2
• ? C ?o A/d (capacitance of a
  parallel- plate capacitor)

-Q
Q
A
B
d
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Capacitors in Series and Parallel Combination

• Symbol

Constant Capacitor
)
Variable Capacitor
27
Capacitors in Series Combination

• Series Combination
• Magnitude of charge Q is the same on all
  plates
• Q1 Q2 Q3 Qn Qtotal
• Vtotal V1 V2 V3 Vn
• 1/Ctotal 1/C1 1/C2 1/C3 1/Cn

28
Capacitors in Parallel Combination

• Parallel Combination
• Potential Difference ?V is the same for all
  capacitors connected in parallel
• V1 V2 V3 Vn Vtotal
• Qtotal Q1 Q2 Q3 Qn
• Ctotal C1 C2 C3 Cn

29
Capacitors in Complex Combinations
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Capacitors with Di-Electrics

• Capacitors with Di-Electric
• A capacitor is always made up of two conductors
  of opposite charge.

31
Capacitors with Di-Electrics (cont.)

• Dielectric
• An insulation material that fills the space
  between the conductors of a capacitor. The
  dielectric increases the original capacitance by
  a factor of k, so the new capacitance is
• Cnew k Cold
• k di-electric constant
• The di-electric also serves to decrease the
  potential difference of the capacitance by a
  factor of 1/k, so
• Vnew 1/k Vold
• The di-electric also serves to change the
  magnetic properties of the space between the
  conductors.

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Capacitors with Di-Electrics (cont.)

• Summary
• Cnew k Cold (new capacitor)
• Qnew Qold (charge does not change in new
  capacitor)
• Vnew 1/k Vold (potential difference in new
  capacitor)

33
Comparison of Parallel-Plate Capacitors with and
without di-electric
Without di-electric Cold ?o A/d (capacitance
of a parallel-plate capacitor in vacuum/air) With
di-electric Cnew k ?o A/d (capacitance of a
parallel-plate capacitor with di-electric in
vacuum/air)
d
A
-Q
Q
34
Capacitors with Di-Electrics (cont.)

• Purposes of a di-eletric
• Serves to separate the plates from making
  contact.
• Decreases the potential difference by a factor of
  1/k. when capacitor is charged, voltage drops to
  Vnew 1/k Vold when di-electric is inserted.

35
Energy stored in a charged Capacitor

• Energy stored in a charged capacitor

V
W Area under curve ½ V Q Energy stored
in capacitor ECapacitor
W E
Q
36
Energy stored in a charged Capacitor (cont.)

• Equations
• EC ½ Q V
• EC ½ C V2
• EC ½ Q2/C