Engineering Physics

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Engineering Physics

• 4. Electric Potential

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Main Topics
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Definition of Work
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Work done by Gravity
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Energy

• Energy is the capacity to do work
• Potential energy energy by virtue of position
  e.g. mgh joules
• Kinetic energy energy by virtue of motion e.g.
  ½ mv2 joules
• Energy can be converted from potential to kinetic
  or visa versa

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Troll Quiz
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Converting PE to KE
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EPE Electric Potential Energy
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Convert EPE to KE
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Electric Potential (voltage) I
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Electric Potential (voltage) II

• We can now relate the work WAB done by the
  electric field when a charge q0 moves from A to B
  to the potential difference VB VA between the
  points.
• VB VA (EPEB/q0) - (EPEA/q0)
• - (WAB/q0)
• Or DV (DEPE/ q0) - (WAB/q0)

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Electric Potential (voltage) III
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Force v Potential
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Correct SI Units
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Worked Example
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Definition of Electric Potential Difference

• The electric potential difference between two
  points B and A is the work done (WAB) per unit
  charge by an external force in taking a positive
  charge from A to B
• VB - VA WAB/Q (UB UA)/Q
• Or in integral form

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Potential at a distance r from a Positive Charge Q
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Calculate the Potential at A B
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Where is the Potential Zero?
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Scalar Field
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The Electric Potential Energy of a Group of
Charges

• Equilateral triangle
• Top charge put in place first
• Left hand charge now added
• Finally, right hand charge is put in place
• Calculate the EPE of the system
• Does the order of placing the charges at the
  three corners matter?
• Could you calculate the potential at the centroid
  of the triangle?

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Troll Quiz
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Equipotential Surfaces
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Equipotential Lines I
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Electric Field Lines
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Equipotential Lines II
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Potential Gradient

• The differential form of the equation opposite
  is -

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Important Points

• The net electric force does no work as a charge
  moves on an equipotential surface.
• The electric field created by any charge or group
  of charges is everywhere perpendicular to the
  associated equipotential surfaces and points in
  the direction of decreasing potential.

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Parallel Plate Capacitor

• Plate separation 3.2cm
• Plate PD is 64 V ( to -)
• Equipotential (light blue) surface PD is 3V (left
  to right)
• Hence, their separation is (3/64)x(0.032) m
• Note VB VA - 64 V

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Troll Quiz
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Capacitors

• For two parallel plates, each holding a charge of
  magnitude Q, it is found experimentally that -
• Q CV
• V is the voltage between the plates.
• C is a constant, and is called the capacitance of
  the device.
• C is measured in farads (F).
• A capacitor is a charge storage device.

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Dielectrics I

• When a dielectric (insulating material) is placed
  between plates the electric field is reduced from
  E0 to E.
• The dielectric constant is given by
• k E0/E

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Dielectrics II

• For a uniform field, from the definition of
  voltage, we have
• V Ed, where d is the plate spacing.
• If the electric field is reduced, then so is the
  voltage.
• If the capacitor is isolated, Q remains
  unchanged, and Q CV, then C must increase to
  balance the equation.

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Some Dielectric Constants
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Proving the Capacitor Formula

• A is area of one plate d the plate spacing. E
  is the electric field V the voltage between the
  plates.
• From definition of voltage E Vd
• From Gauss Law E s/e0 Q/Ae0
• Hence, Q (e0A/d)V or Q CV
• Hence, C e0A/d farads
• When a dielectric is present, replace e0 by ke0
  in the formula for C. That applies also to
  Coulombs law.

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Worked Example - CJ 6ed 19.5
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Energy Storage in a Capacitor I

• The electric force (felt by a charge dq) between
  plates is E(dq) newtons.
• The work done in bringing this small charge from
  the negative plate to the positive plate is
  E(dq)d joules.
• E increases as charge is added in small amounts,
  dq, to the positive plate. E is a function of q,
  the charge on the plate at any time.

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Energy Storage in a Capacitor II

• V q/C Ed, d is the plate spacing V increases
  as charge is added to the positive plate.
• Hence, Ed q/C - substitute for Ed in (3.)
  above and integrate for o to Q Q is the final
  charge on the positive plate and V is the final
  voltage between the plates.
• Energy(work done in charging) is
• ½ (Q2/C) ½ CV2

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Energy Density

• The volume of the capacitor is Ad m3.
• The energy per unit volume for a parallel plate
  capacitor is as follows
• Energy density ½ (ke0A/d)(Ed)2/Ad
• ½ ke0E2
• This expression holds for any electric field
  strength, not just between the plates of a
  capacitor.

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Troll Quiz