PHY 184 Week 5 Notes
PHY 184 Week 5 Notes PHY 184
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This 6 page Class Notes was uploaded by Cameron Blochwitz on Saturday September 24, 2016. The Class Notes belongs to PHY 184 at Michigan State University taught by Oscar Naviliat Cuncic in Fall 2016. Since its upload, it has received 3 views. For similar materials see Physics for Scientists and Engineers II in Physics at Michigan State University.
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Date Created: 09/24/16
PHY 184 Week 5 Notes-Capacitors 9/27-9/30 Capacitors o Device that can store energy Many different uses o Camera Flash o Touch Screens Essential in Electronics Capacitor consist of two separated conductors with an insulating layer o Can be rolled up with another insulator to increase the surface and storage A simple geometry of a capacitor each with area A and distance d in a vacuum with charge on one plate of +q and a charge on the other of -q The surface of the conductor is an equipotential surface o Equipotential lines are close to each other near the plate and far away outside of the plates Electric field lines are perpendicular to equipotential lines o Almost uniform between the plates o Far from the plates it acts as a dipole Field outside of the plates is called the fringe field Capacitance The main property of a Capacitor is its Capacitance Capacitance is given by q o C=| | ∆V Capacitance is an intrinsic value of the capacitor o It cannot change Capacitance shows how much charge can be stored at a given Voltage ∆V o q=C ¿) Farad The unit of capacitance is called the Farad 1C o 1F= 1V 1 Farad is a very large capacitance o Most Capacitors range from pF to μF Finite Parallel Plates We already know that σ q o E= ε = Aε o 0 The voltage difference between the plates is ∆ V=−∫Eds o From the negative to positive plate −Eds =¿Ed d o ∆V=− ∫ 0 Capacitance of Parallel Plates is therefore Aε C= q = 0 o ∆V| d Cylindrical Capacitor Consider a capacitor constructed by 2 co-axial conducting cylinders of length L inner radius r a1d outer radius r 2 o Inner cylinder has charge -q and outer cylinder has charge +q From symmetry electric field is radial ∯ E ∙dA=−E ∯ dA=−E 2πrL = −q ε0 q o E= ε 2πrL 0 Voltage difference is f r2 2 q q r o ∆ V=−∫E∙ds=− E∫r= ∫ dr= ln 2 i r1 1ε02πrL 0 2πL r1 Capacitance is then q q ε 2πL C= = = 0 o |∆V q r2 r2 ln ln ε02πL r1 r1 Spherical Capacitor Capacitor made of 2 concentric spheres o Inner sphere charged to -q with radius r 1 o Outer sphere charged to +q with radius r 2 Electric Field is radial and uniform Applying Gauss Law E ∙dA=EA=E(4πr )=2 q o ∯ ε0 q E= 2 4πε 0 Voltage difference is f 2 2 q −q 1 1 o ∆ V=− ∫∙ds=− Ed∫= ∫ 2dr= 4πε r( )− r i 1 14πε 0 0 1 2 Capacitance r r C=4πε 0 1 2 o r2−r1 Capacitance of a Single Sphere The capacitance of a single sphere is obtained from the above result with the outer sphere having an infinite radius C=4πε R o 0 Potential is then q o ∆ V=k r Electric Properties of a Capacitor These configurations use a vacuum to separate the plates o ε d0termines the electrical properties Dielectric Materials 2 layers of a capacitor are separated by and insulating material that is dielectric o 2 kinds of dielectric material, polar and non-polar Polar materials have a permanent dipole moment These dipoles are randomly oriented Electric fields will make them all point the same way Non-polar materials have no permanent dipole moment Can be polarized to align in an electric field o Smaller effect than polar These materials serve several purposes o Provides a way to maintain separation o Provides insulation o Increases the capacitance o Allows capacitor to hold a higher voltage Electric Field in a Dielectric In a dielectric, the Field d resulting from the aligned dipole will reduce E ET=E−E d o Capacitor with Dielectrics Placing a dielectric has the effect of lowering the electric field allowing more charge to be stored It will increase the capacitance of the capacitor by a dielectric constant κ. C=κC o air κ ≡1,κ ≡1 o vaccum air Parallel Plate Capacitor Placing a dielectric between the plates E=E a=r q = q o κ κε0A εA Electric permittivity o ε=κε 0 Making Capacitance εA o C= d Work to Charge a Capacitor When a battery charges a capacitor charge accumulates on the capacitor We have to work to put charges on something already charged Work is therefore dW=∆V ' dq' o If there is a capacitance C dW=∆V dq =' q'dq' o C Total work is then qq' 1 q o Wt=∫dW= ∫ dq'= 0C 2 C Energy stored in the Capacitor is then given by 1 q2 1 2 1 o U= = CV = qV 2 C 2 2 Electric Energy Density, u, is defined by potential energy per volume For a Parallel Plate Capacitor U CV 2 1 2 o u= Ad= 2Ad =2ε 0 Circuits with Capacitors Electric Circuits are a set of elements connected with wires These need some form of power source Wires Wires are represented by a line o All points of the wire are at the same potential Equivalent to a point Charging and Discharging a Capacitor When connected to a battery, current will flow until the capacitor is charged Combination of Capacitors Parallel o Potential is the same across all the capacitors V=V =1 =…2V n o Charges are given by q=q +q +…+q =(C +C +…C )V 1 2 n 1 2 n Series o Charge is the same across the series V= q o c eq 1 1 1 1 = + +…+ C eq C 1 C 2 C n
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