Circuit Elements Read: Chapter 20 Capacitors

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1 Read: Chapter 20 Capacitors capacitor circuit symbol A capacitor, in its simplest form, consists of two conducting plates separated by a small gap. Current cannot flow through a capacitor. A capacitor is capable of storing charge, so can have a potential difference across its plates. A fully charged capacitor may be discharged by connecting it to a closed circuit. As charge leaves the capacitor, E decreases in the circuit, causing current to decrease. Once the capacitor is fully discharged, E = 0 and current stops. E + E d E dt < 0 bulb Question: Using the analysis from Chapter 19, explain why the electric field decreases throughout the circuit as the capacitor discharges. 1

2 Capacitors A capacitor may be charged by connecting it to a source of emf (e.g., a battery). As charge collects on the capacitor, E decreases throughout the circuit until the capacitor becomes fully charged and E and I drop to 0. E + + E d E dt < 0 battery Question: Use the analysis of Chapter 19 to explain why E in the circuit decreases as the capacitor charges. The rate of charging (or discharging) of a capacitor is dependent upon the current in the circuit large current fast charging, small current slow charging. Exercise: Give a physical explanation for the dependence of charging rate on current magnitude. 2

3 Capacitors In analyzing current in a circuit with a capacitor, the capacitor may be thought of as a single current node the current entering one plate must be equal in magnitude to the current leaving the other. Capacitance C is a the physical property of a capacitor that determines how much charge the capacitor can store at a given potential difference V ; it is defined by Q = C V. The value of capacitance is primarily determined by the geometry of the capacitor; for a parallel-plate capacitor, capacitance is given by C = ɛ 0 (A/s), where A is the surface area of a single plate and s is the separation distance between plates. 3

4 Capacitors In circuit analysis, it is convenient to sometimes define an equivalent capacitance C eq. Given some configuration of multiple capacitors, its equivalent capacitance is the value of C that a single capacitor would have if substituted for the configuration. V For capacitors connected in series, we can calculate an equivalent capacitance C eq that is defined by 1 C eq = 1 C C 2 +. C 1 C 2 V C 1 C 2 For capacitors connected in parallel, the equivalent capacitance C eq is defined by C eq = C 1 + C

5 Resistors Conductivity σ is a material property that determines the size of the current in the material at a particular value of applied electric field E. It is defined by σ q nu, where n is the density of charge carriers and u is their mobility. Current density J is obtained by dividing current I by the cross-sectional area A. Using I = q nuae, J is defined by J σ E. Resistance R is a physical property that combines material and geometric elements. The definition of resistance is R L σa, where L is the length of the circuit element and A its cross-sectional area. The unit of resistance is ohms: Ω = V/A. 5

6 Resistors Ohm s Law is given by V = IR, and is valid for materials whose resistance R is independent of current. Materials that obey Ohm s Law are said to be ohmic materials. Question: Why are capacitors and batteries not ohmic? resistor circuit element A resistor is a circuit element that obeys Ohm s Law. Resistors may be used to control voltage or current within a circuit. 6

7 Resistors As with capacitors, we may define an equivalent resitance R eq for some configuration of multiple resistors. V For resistors connected in series, equivalent resistance R eq is defined by R eq = R 1 + R 2 +. R 1 R 2 V R 1 R 2 For resistors connected in parallel, the equivalent resistance R eq is defined by 1 R eq = 1 R R

8 Work and Power The power consumed in any circuit component is given by P = I V. Exercise: (a) Calculate the number of electrons that flow past A each second in the steady-state. B 1.5 V 1.5 V A (b) Draw a graph of potential vs. circuit location, using the labeled points as points on the x-axis. (c) What is the power output of the battery? D 40 Ω C 10 Ω B 8

9 Batteries emf r int A real battery may be modeled as an ideal battery coupled to an internal resistance r int : V battery = emf r int I. Question: Without internal resistance, what would the current be in a circuit consisting only of a battery and an ideal (R = 0) wire? Question: The internal resistance of a battery places an upper limit on what physical quantity? 9

10 RC Circuits emf R Q +Q I A simple RC circuit consists of a battery, a resistor, and a capacitor connected in series. The loop equation for the simple RC circuit is emf IR Q/C = 0 C In its steady state, the RC circuit has I f = 0 and the capacitor is fully charged with charge Q = (emf)c. Most generally, the RC circuit has emf Q(t)/C I(t) =. R To find the charge Q on the capacitor at some arbitrary time t, we must solve the differential equation I(t) = dq dt = emf Q(t)/C R. 10

11 RC Circuits Q(coulombs) Q 1 e t/rc The solution to the differential equation for Q(t) is Q(t) = C(emf)(1 e t/rc ), plotted at left in arbitrary units. As t, Q C(emf) as expected. t(seconds) I(amperes) I(t) can be obtained by differentiation of Q(t), since I = dq/dt. Doing so, we obtain ( ) emf I(t) = e t/rc, R again plotted in arbitrary units. As t, I 0 as expected. t(seconds) I e t/rc 11

12 RC Circuits The time constant τ is defined as the time it takes for the exponential term to reach e 1 ; for RC circuits, τ = RC. The time constant provides a rough guide to the rate at which exponential processes evolve. At right are some integer values of τ on the I vs. t plot. I(amperes) t = τ t(seconds) t = 2τ I e t/rc 12

Problem Solving 8: RC and LR Circuits

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Problem Solving 8: RC and LR Circuits Section Table and Group (e.g. L04 3C ) Names Hand in one copy per group at the end of the Friday Problem