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Chapter 19: Electric Potential &

Potential Energy

Brent Royuk Phys-112

Concordia University

Terminology •  Two Different Quantities:

–  Electric Potential and Electric Potential Energy •  Electric Potential = Voltage •  Note: We will start by considering a point

charge, section 18-3.

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Electric Potential Energy •  Consider two point charges separated by a

distance r. The energy of this system is

•  To derive this, you need to integrate work using Coulomb’s Law.

•  Potential energies are always defined relatively. Where is U = 0 for this system?

•  What is negative energy? •  This is a scalar quantity. •  The Superposition Principle applies. •  We are most often interested in changes and

differences, rather than absolutes.

€

U =kqoq

r

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Electric Potential •  Definition:

–  This is called the electric potential (which shouldn’t be confused with electric potential energy), the potential, or the voltage.

–  Remember: potential is energy per charge. •  Units

–  In MKS, energy/charge = Joule/Coulomb = 1 volt (V)

•  In everyday life, what’s relevant about this infinity stuff? Nothing, really. –  Potentials tend to be differences. One

commonly chosen zero: the earth.

€

V =Uqo

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Comparisons •  An Analogy

–  Coulomb Force --> Electric Field (Force per charge), as

–  Electric Potential Energy --> Electric Potential (Energy per charge)

•  How is electric potential energy similar to gravitational potential energy?

•  Potential in this chapter compared to future chapters.

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Electric Potential •  For a point charge,

€

V =Uqo

=kqqo

rqo

=kqr

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Electric Potential Examples •  A battery-powered lantern is switched on for 5.0

minutes. During this time, electrons with total charge -8.0 x 102 C flow through the lamp; 9600 J of electric potential energy is converted to light and heat. Through what potential difference do the electrons move?

•  Find the energy given to an electron accelerated through a potential difference of 50 V. –  a) The electron volt (eV)

•  An electron is brought to a spot that is 12 cm from a point charge of –2.5 µC. As the electron is repelled away, to what speed will it finally accelerate?

•  Find the electric field and potential at the center of a square for positive and negative charges. –  What do positive and negative voltages mean? –  E-field lines point in the direction of decreasing V.

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Electric Potential Examples •  How much work is required to assemble the

charge configuration below?

2  3

1 4

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Electric Potential Examples •  Consider the three charges shown in the figure

below. How much work must be done to move the +2.7 mC charge to infinity?

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Potential in a Uniform Field •  Let’s let an electric field do some work

as we move a test-charge against the field:

•  The work done by the field is: W = -qoEd

•  Assuming we start at the U = 0 point, we get U = -W = qoEd

•  Signs? See next slide. •  Using the definition of the potential we

get: V = Ed

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Potential in a Uniform Field •  Sign considerations: •  Work done by the field is negative, which

makes the potential energy positive (useful). –  Compare with gravity:

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Potential in a Uniform Field

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Potential in a Uniform Field •  Example: A uniform field is

established by connecting the plates of a parallel-plate capacitor to a 12-V battery. a) If the plates are separated by 0.75 cm, what is the magnitude of the electric field in the capacitor? b) A charge of +6.24 µC moves from the positive plate to the negative plate. How much does its electric potential energy change?

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Equipotential Surfaces •  An equipotential surface has the

same potential at every point on the surface.

•  Equipotential surfaces are perpendicular to electric field lines. –  The electric field is the gradient of

the equipotential surfaces.

•  How are equipotential lines oriented to the surface of a conductor?

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Equipotential Surfaces

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Equipotential Surfaces •  Comparative examples:

–  Isobars on a weather map. –  Elevation lines on a topographic map.

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Capacitors •  A plate capacitor •  It takes energy to charge the plates

–  Easy at first, then harder

•  Q = CV –  C is the capacitance –  Bigger C means more charge per volt, bigger charge

storage device –  1 farad (F) = 1 coulomb/volt

•  –  εo = 8.85 x 10-12 C2/Nm2 (permittivity of free space) –  Connect with k

•  What area plate separated by a gap of 0.10 mm would create a capacitance of 1.0 F?

€

C =εoAd

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Capacitors in Circuits •  Series

–  Charge is same on all capacitors –  Voltage drops across the capacitors –  So V = V1 + V2 + V3 +... –  Since V = Q/C, –  Therefore:

•  Parallel –  The voltage is the same across all capacitors.

Different amounts of charge collect on each capacitor

–  Q = Q1 + Q2 + Q3 +... –  Q = CV, so CeqV = C1V + C2V + C3V + ... –  Generally,

€

QCeq

=QC1

+QC2

+QC3

+ ...

€

1Ceq

=1C1

+1C2

+1C3

+ ...

€

Ceq = C1 + C2 + C3 + ...

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Dielectrics •  In real life, capacitor plates are not

naked, the gap is filled with a dielectric material –  Dielectrics are insulators. –  Keeps plates separated, easier to build. –  Also increases the capacitance

•  The dielectric constant –  Isolated capacitor: insert dielectric, E is

reduced by 1/κ •  κ = the dielectric constant •  C = κCo

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Dielectrics

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Electrical Energy Storage •  Graph V vs. q:

•  What is the area under the curve?

V

Q

Slope = 1/C

€

U =12

QV =Q2

2C=

12

CV 2

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Electrical Energy Storage •  A defibrillator is used to deliver 200 J of

energy to a patient’s heart by charging a bank of capacitors to 750 volts. What is the capacitance of the defibrillator?

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