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 kq q U = o r • 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.

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1 Electric Potential U • Definition: V = q o – 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. 4

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, U kqq kq V = = o = q rq r o o

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2 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?

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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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3 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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4 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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5 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 ε A • C = o d -12 2 2 – εo = 8.85 x 10 C /Nm (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?

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Capacitors in Circuits • Series – Charge is same on all capacitors – Voltage drops across the capacitors

– So V = V1 + V2 + V3 +... Q Q Q Q = + + + ... – Since V = Q/C, C C C C eq 1 2 3 – Therefore: 1 1 1 1 = + + + ... € C C C C • Parallel eq 1 2 3 – 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, C C C C ... eq = 1 + 2 + 3 +

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6 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: V Slope = 1/C

Q • What is the area under the curve?

1 Q2 1 U = QV = = CV 2 2 2C 2

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7 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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