1January 30, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 13January 30, 2005 Physics for Scientists&Engineers 2 2ReviewReview Capacitors are devices that can store electrical energy The definition of capacitance is•C is the capacitance•q is the the charge on the capacitor• +q on one plate• -q on the other plate•V is the voltage across the plates The unit of capacitance is the farad (abbreviated F)C =qV1 F =1 C1 VJanuary 30, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2) The capacitance C of a parallel plate capacitor is given by Where•A is the area of each plate•d is the distance between the plates•ε0 is the electric permittivity of free space Note that this result for the capacitance of a parallel plate capacitor dependsonly on the geometry of the capacitor and not on the amount of charge or thevoltage across the capacitorC =!0Ad+q!qJanuary 30, 2005 Physics for Scientists&Engineers 2 4Review (3)Review (3) The capacitance of a cylindrical capacitor is•L is the length of the cylinder•r1 is the radius of the inner cylinder•r2 is the radius of the outer cylinder•λ is the charge per unit length for both cylinders• +q is the charge on the inner cylinder• -q is the charge on the outer cylinderC =2!"0Lln r2/ r1( )2January 30, 2005 Physics for Scientists&Engineers 2 5Spherical CapacitorSpherical Capacitor Consider a spherical capacitor formed by two concentric conductingspheres with radii r1 and r2January 30, 2005 Physics for Scientists&Engineers 2 6Spherical Capacitor (2)Spherical Capacitor (2) Let’s assume that the inner sphere has charge +q and the outer spherehas charge -q The electric field is perpendicular to the surface of both spheres andpoints radially outwardJanuary 30, 2005 Physics for Scientists&Engineers 2 7Spherical Capacitor (3)Spherical Capacitor (3) To calculate the electric field, we use a Gaussian surfaceconsisting of a concentric sphere of radius r such that r1 < r < r2 The electric field is always perpendicular to the Gaussian surface so Which reduces to !0!Eid!A =!0EA =!0E 4"r2( )= q"#E =q4!"0r2January 30, 2005 Physics for Scientists&Engineers 2 8Spherical CapacitorSpherical Capacitor (4)(4) To get the electric potential we follow a method similar to the one weused for the cylindrical capacitor and integrate from the negativelycharged sphere to the positively charged sphere Using the definition of capacitance we get The capacitance of a spherical capacitor is thenV = Edrr2r1!=q4"#0r2drr2r1!=q4"#01r1$1r2%&'()*C =qV=qq4!"01r1#1r2$%&'()=4!"01r1#1r2$%&'()C = 4!"0r1r2r2# r13January 30, 2005 Physics for Scientists&Engineers 2 9Capacitance of an Isolated SphereCapacitance of an Isolated Sphere We can obtain the capacitance of a single conductingsphere by taking our result for a spherical capacitor andmoving the outer spherical conductor infinitely far away Using our result for a spherical capacitor withr2 = ∞ and r1 = R we getC = 4!"0RJanuary 30, 2005 Physics for Scientists&Engineers 2 10Capacitors in CircuitsCapacitors in Circuits A circuit is a set of electrical devices connectedwith conducting wires Capacitors can be wired together in circuits inparallel or series• Capacitors in circuits connected by wires such that thepositively charged plates are connected together andthe negatively charged plates are connected together,are connected in parallel• Capacitors wired together such that the positivelycharge plate of one capacitor is connected to thenegatively charged plate of the next capacitor areconnected in series.January 30, 2005 Physics for Scientists&Engineers 2 11Capacitors in ParallelCapacitors in Parallel Consider an electrical circuit with three capacitors wired inparallel Each of three capacitors has one plate wired directly tothe positive terminal of a battery with voltage V and oneplate wired directly to the negative terminal The potential difference V across each capacitor is thesame We can write the charge on each capacitor asq1= C1V q2= C2V q3= C3VJanuary 30, 2005 Physics for Scientists&Engineers 2 12Capacitors in Parallel (2)Capacitors in Parallel (2) We can consider the three capacitors as one equivalentcapacitor Ceq that holds a total charge q given by We can now define Ceq as A general result for n capacitors in parallel is If we can identify capacitors in a circuit that are wired inparallel, we can replace them with an equivalent capacitanceq = q1+ q2+ q3= C1V + C2V + C3V = C1+ C2+ C3( )VCeq= C1+ C2+ C3Ceq= Cii =1n!q = CeqV4January 30, 2005 Physics for Scientists&Engineers 2 13Capacitors in SeriesCapacitors in Series Consider a circuit with three capacitors wired in series The positively charged plate of C1 is connected to thepositive terminal of the battery The negatively charge plate of C1 is connected to thepositively charged plate of C2 The negatively charged plate of C2 is connected to thepositively charge plate of C3 The negatively charge plate of C3 is connected to thenegative terminal of the battery The battery produces an equal charge q on each capacitor because thebattery induces a positive charge on the positive place of C1, whichinduces a negative charge on the opposite plate of C1, which induces apositive charge on C2, etc.January 30, 2005 Physics for Scientists&Engineers 2 14Capacitors in Series (2)Capacitors in Series (2) Knowing that the charge is the same on all three capacitorswe can write We can express an equivalent capacitance Ceq as We can generalize to n capacitors in series If we can identify capacitors in a circuit that are wired in series, wecan replace them with an equivalent capacitanceV = V1+ V2+ V3=qC1+qC2+qC3= q1C1+1C2+1C3!"#$%&V =qCeq1Ceq=1C1+1C2+1C31Ceq=1Cii =1n!January 30, 2005 Physics for Scientists&Engineers 2 15Example - System of CapacitorsExample - System of Capacitors Let’s analyze a system of five capacitors If each capacitor has a capacitance of 5 nF, whatis the capacitance of this system of capacitors?January 30, 2005 Physics for Scientists&Engineers 2 16System of Capacitors (2)System of Capacitors (2) We can see that C1 and C2 are in parallel and that C3 is alsoin parallel with C1 and C2 We can define And make a new drawingC123= Cii =13!= C1+ C2+ C35January 30, 2005 Physics for Scientists&Engineers 2 17System of Capacitors
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