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UT Arlington PHYS 1444 - Potential from System of Charges, Capacitors and Capacitance

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Tuesday, Feb. 15, 2011 1PHYS 1444-02 Dr. Andrew BrandtPHYS 1444 – Section 02Lecture #7• Chapter 23:• Chapter 24 Tuesday Feb 15, 2011Dr. Andrew Brandt• Potential from System of Charges• Capacitors and Capacitance• HW3 due Weds at 9pmTuesday, Feb. 15, 2011 2PHYS 1444-02 Dr. Andrew BrandtPotential due to Point Charges• E field due to a point charge Q at a distance r? • Electric potential due to the field E for moving from point rato rbaway from the charge Q isbaVVbarrE dlE2014Qr2Qkr20ˆˆ4barrQrrdrr2014barrQdrr0114baQrr3Potential due to Electric Dipolesr=lcosVV014iiaQr014QQr r r0114Qr r r04 ( )Qrr r r20cos4Qlr201 cos4pVrV due to dipole a distance r from the dipoleTuesday, Feb. 15, 2011 PHYS 1444-02 Dr. Andrew BrandtTuesday, Feb. 15, 2011 4PHYS 1444-02 Dr. Andrew BrandtE Determined from V• Potential difference between two points is• So we can write– What are dV and El?• dV is the infinitesimal potential difference between two points separated by the distance dl• Elis the field component along the direction of dl.baVVbaE dlldVEdlEVi j k Vx y zTuesday, Feb. 15, 2011 5PHYS 1444-02 Dr. Andrew BrandtElectrostatic Potential Energy• Consider a point charge q that is moved between points aand b where the electrostatic potentials due to other charges are Vaand Vb• The change in electrostatic potential energy of q is• Now what is the electrostatic potential energy of a configuration of charges?– Let’s choose V=0 at r=– If there are no other charges around, a single point charge Q1in isolation has no potential energy and feels no electric forceUbaUUbaqV VbaqVTuesday, Feb. 15, 2011 6PHYS 1444-02 Dr. Andrew BrandtElectrostatic Potential Energy; Two charges• If a second point charge Q2is brought close to Q1at the distance r12, the potential due to Q1at the position of Q2is• The potential energy of the two charges relative to V=0 at r= is– This is the work that needs to be done by an external force to bring Q2from infinity to a distance r12from Q1.– It is also a negative of the work needed to separate them to infinity.V10 1214Qr2U Q V120 1214QQrTuesday, Feb. 15, 2011 7PHYS 1444-02 Dr. Andrew BrandtElectrostatic Potential Energy; Three Charges• So what do we do for three charges? • Work is needed to bring all three charges together– Work needed to bring Q1to a certain place without the presence of any charge is 0.– Work needed to bring Q2to a distance to Q1is– Work need to bring Q3to a distance to Q1and Q2is• So the total electrostatic potential of the three charge system is12U3UU120 1214QQr13U130 1314QQr230 2314QQr23U12 13 23U U U132312012 13 231 0 at 4QQQQQQVrr r rTuesday, Feb. 15, 2011 8PHYS 1444-02 Dr. Andrew BrandtElectrostatic Potential Energy: electron Volt• What is the unit of electrostatic potential energy?– Joules• Joules is a very large unit in dealing with electrons, atoms or molecules• For convenience a new unit, electron volt (eV), is defined– 1 eV is defined as the energy acquired by a particle carrying the charge equal to that of an electron (q=e) when it moves across a potential difference of 1V.– How many Joules is 1 eV then?• eV however is not a standard SI unit. You must convert the energy to Joules for computations.1eV191.610 1CV191.6 10 JTuesday, Feb. 15, 2011 9PHYS 1444-02 Dr. Andrew BrandtCapacitors (or Condensers)• What is a capacitor?– A device that can store electric charge without letting the charge flow• What does it consist of?– Usually consists of two oppositely charged conducting objects (plates or sheets) placed near each other without touching– Why can’t they touch each other?• The charges will neutralize each other• Can you give some examples?– Camera flash, surge protectors, computer keyboard, binary circuits…• How is a capacitor different than a battery?– Battery provides potential difference by storing energy (usually chemical energy) while the capacitor stores charge but very little energy.10Capacitors• A simple capacitor consists of a pair of parallel plates of area A separated by a distance d.– A cylindrical capacitors are essentially parallel plates wrapped around as a cylinder.• How would you draw symbols for a capacitor and a battery?– Capacitor -||-– Battery (+) -|i- (-)Circuit DiagramTuesday, Feb. 15, 2011Tuesday, Feb. 15, 2011 11PHYS 1444-02 Dr. Andrew Brandt• What do you think will happen if a battery is connected (voltage is applied) to a capacitor?– The capacitor gets charged quickly, one plate positive and the other negative with an equal amount. of charge• Each battery terminal, the wires and the plates are conductors. What does this mean?– All conductors are at the same potential. – the full battery voltage is applied across the capacitor plates.• So for a given capacitor, the amount of charge stored in the capacitor is proportional to the potential difference Vbabetween the plates. How would you write this formula?– C is a proportionality constant, called capacitance of the device.– What is the unit? CapacitorsbaQ CVC/V or Farad (F)C is a property of a capacitor so does not depend on Q or V.Normally use F or pF.Tuesday, Feb. 15, 2011 12PHYS 1444-02 Dr. Andrew BrandtDetermination of Capacitance• C can be determined analytically for capacitors w/ simple geometry and air in between.• Let’s consider a parallel plate capacitor.– Plates have area A each and separated by d.• d is smaller than the length, so E is uniform.– For parallel plates E=0, where is the surface charge density.• E and V are related• Since we take the integral from the lower potential point a to the higher potential point b along the field line, we obtain•• So from the formula:– What do you notice?baVbaV00baAQQCV Qd A dC only depends on the area (A) and the separation (d) of the plates and the permittivity of the medium between them.baVVcos180baEdlbaEdl0badl0baQdlA0baQdlA0QbaA0QdAbaE dlTuesday, Feb. 15, 2011 13PHYS 1444-02 Dr. Andrew BrandtExample 24 – 1Capacitor calculations: (a) Calculate the capacitance of a capacitor whose plates are 20 cm x 3.0 cm and are separated by a 1.0 mm air gap. (b) What is the charge on each plate if the capacitor is connected to a 12 V battery? (c) What is the electric field between the plates? (d) Estimate the area of the plates needed to achieve a capacitance of 1F, given the same air gap. (a) Using the formula for a parallel plate capacitor, we obtain 0ACd(b) From


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UT Arlington PHYS 1444 - Potential from System of Charges, Capacitors and Capacitance

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