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2054Lecture11

# 2054Lecture11 - Capacitance Resistance& Kirchhoff’s...

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Unformatted text preview: Capacitance, Resistance & Kirchhoff’s Laws Kirchhoff’s Topics of Discussion Topics Definition of Capacitance Types of Capacitors Dielectrics & Capacitors Capacitors in Series & Parallel Electric Currents Resistance & Resistivity Ohm’s Law Resistors in Series & Parallel Electromotive Force Kirchhoff’s Laws & Conventions Definition of Capacitance Definition Given two conductors, we define the Given capacitance C to be a measure of the capacitance charge Q stored on either conductor that produce a potential difference V between the conductors. It is given by Q C= V where the SI unit of capacitance is the Farad 1F=1C/V. 1F=1C/V Types of Capacitors Types Let the area of a plate be A and let d be the Let distance between the plates. Then the capacitance of a parallel-plate capacitor is parallel-plate C=ε0A/d. C= Let a and b be the inner and outer radius of two Let concentric spheres. Then the capacitance of a spherical capacitor is C=4πε0ab/ (b−a) . C=4 ab/ Let R be the radius of an isolated spherical Let capacitor. Then the capacitance is C=4πε0R. C=4 Let a and b be the inner and outer radius of two Let concentric cylinders both of length L. Then the Then capacitance of a cylindrical capacitor is C=2πε L/ln(b/a) . Parallel-Plate Capacitor with a Dielectric Dielectric Let a parallel-plate capacitor consists of two Let conductors of area A placed a distance d from each other with an insulating material, called a dielectric, iinserted between the conductors. The dielectric nserted capacitance is given by capacitance C=κε0 A/d A/d where κ is the dielectric constant of the where dielectric insulating material. insulating Microscopic Description of a Dielectric We view a dielectric as a collection of molecules. A molecule is said to be polarized when there is polarized a separation between the “center of gravity” of its negative and positive charges. its An asymmetric molecule has a permanent An asymmetric polarization. polarization. A symmetric molecule has no permanent symmetric polarization but can be given a polarization, called the induced polarization, by placing it in induced by an external electric field. an Capacitors in Series & Parallel Capacitors The equivalent capacitance C for two The capacitors C1 and C2 connected in series series is given by is 1 1 1 = + C C1 C 2 The equivalent capacitance C for two The capacitors C1 and C2 connected in parallel parallel is given by is C = C1 + C 2 Electric Current Electric The time rate change Δt at which charges q The pass through a cross-sectional area A is called the electric current, and is given by electric where current is measured in the ampere where A=C/s. Conventional current flows in the A=C/s direction in which positive charges flow, even though the actual charge carries are negative and flow in the opposite direction. and ∆q I= ∆t Flow of Electric Current Flow Recall that conventional current is the Recall direction in which positive charge carriers flow. Electrons are the charge carriers for electric current. They move in random directions with a drift velocity vd given by drift I vd = . nAe Resistance & Resistivity Resistance The resistance R of a conductor is proportional The resistance to its length L and inversely proportional to its cross-sectional area A. L R∝ A The unit of resistance R is the ohm Ω, and is The given by given L R= ρ A where ρ is the resistivity of the conductor in Ωm. resistivity Ohm’s Law Ohm’s The resistance R remains constant over a The wide range of applied voltages V and currents I. V = IR Resistors in Series & Parallel Resistors The equivalent resistance R for two The resistors R1 and R2 connected in series is given by given R = R1 + R2 The equivalent resistance R for two The resistors R1 and R2 connected in parallel is given by given 1 1 1 = + R R1 R2 Power Power The amount of power dissipation of a The resistor R held at an applied voltage V or current I is given by V P= I R= R 2 2 Electromotive Force (emf) Electromotive A battery source that maintains the current I in a battery closed loop is called the source of emf E . source The terminal voltage of a battery is given by ΔV The terminal = E − Ir, where the internal resistor r is the Ir where resistance of the battery. resistance The load resistor R is the resistance of applied The load to the battery. The source of emf is then given emf by E = IR + Ir. Ir Kirchhoff’s Laws Kirchhoff’s The sum of currents entering a junction must The equal the sum of the currents leaving that junction. junction. ∑ I in = ∑ I out Around any closed circuit loop, the sum of the Around potential drops must equal the sum of the potential rises. potential ∑ Vdrops = ∑ Vrises Another Form of Kirchhoff’s Laws Another The sum of currents entering or leaving any The junction must vanish. junction junction ∑ I=0 The sum of all potential drops and potential rises The around any closed circuit loop must vanish. around ∑ V=0 loop Conventions Conventions Junctions: Assign a direction to the currents entering and leaving a junction. Current entering a junction is assigned a positive sign (+). Current leaving a junction is assigned a negative sign (-). is Loops: Assign a direction to the current in a loop. Across a battery, the emf is assigned a positive sign (+) emf if direction of the loop agrees in going from the negative to the positive terminal of the battery. Across a resistor, the emf is assigned a negative sign (-) if the direction of emf the loop agrees with the direction of the junction current that contains the resistor. Applying the opposite convention yields the same result. RC Circuits RC Let us consider an RC circuit that is Let RC composed of an resistor R and a capacitor C connected in series to an emf source E . emf RC Circuits, cont. RC When a capacitor in an RC circuit is charging, When the charge q(t) on the capacitor at the time t is given by given q ( t ) = Q (1 − e − t /τ ) where τ is called the RC time constant and Q is where RC time the total charge stored by the capacitor. the RC Circuits, cont. RC When a capacitor in an RC circuit is discharging, When the charge q(t) on the capacitor at the time t is given by given q ( t ) = Qe − t /τ where τ is called the RC time constant and Q is where RC time the total charge stored by the capacitor. the ...
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