PHY 1214 General Physics II Lecture 14 Grounding, RC Circuits June 27, 2005 Weldon J. Wilson Professor of Physics & Engineering Howell Hall 221H wwilson@ucok.edu
Lecture Schedule (Weeks 4-6) We are here.
Combine R+C Circuits Gives time dependence Current is not constant I(t) Charge is not constant q(t) Used for timing Pacemaker Intermittent windshield wipers Models nervous system include R, C
Capacitors: Early and Late Initially, when a switch closes there is a potential difference of 0 across an uncharged capacitor. Ultimately, the capacitor reaches its maximum charge and there is no current flow through the capacitor. Therefore, at t=0 the capacitor behaves like a short circuit (R=0), and at t= the capacitor behaves like at open circuit (R= ). Example: 100 V 12.5 A 2.5 A 10 A 0 V Circuit at t=0 at t= Calculate initial currents. Calculate final potentials.
Capacitor Rules of Thumb Initially uncharged capacitor: acts like a wire (short circuit) at t = 0 acts like an open circuit (broken wire) as t Initially charged capacitor: acts like a battery at t = 0
RC Circuits-Discharging
RC Circuits-Discharging a Capacitor How does a capacitor behave? Kirchhoff s loop rule: +IR+q/C=0 Discharging a capacitor. Figure (a) and (b) show the circuit and the charge versus time, respectively. The charge is not removed instantaneously out of to the capacitor.
Discharging a capacitor, cont. Charge: q=q e -t/(rc) Voltage: V=V 0 e -t/(rc) Current: I = I 0 e -t/(rc) I = I 0 e -t/τ Q is the charge on C at t=0 V 0 is the voltage across C at t=0 I 0 is the current thru R at t=0 Fig. 18.17, p.568 Slide 30 τ = RC is called the time constant; after this time the charge decreases to Q/e e =Q/2.718=0.368 0.368Q.
RC Exponential Decay Q( t) Q e RC = 0 t Q t t 0 RC RC I( t) = e = I0e RC 1/ e = 1/ 2.71828 = 0.367879
The switch has been in position a for a long time. It is changed to position b at t=0. What are the charge on the capacitor and the current through the resistor at t=5.0 µs? Example: Exponential Decay in a RC Circuit τ = RC = Ω = µ -6 (10 )(1.0 10 F) 10 s Q C V C -6-6 0 = = (1.0 10 F)(9.0 V) = 9.0 10 C Q µ µ t / RC -6 (5 s) = Q0e = (9.0 10 C) exp(.5) = 5.5 C I 0 Q RC (10 µ s) -6 0 (9.0 10 C) = = = 0.90 A I µ I e t / RC (5 s) = 0 = (0.90 A) exp(.5) = 0.55 A
Charging Capacitor in an RC Circuit The capacitor does not charge instantaneously. Using Kirchhoff s Voltage Rule V 0 IR q C = 0 or V 0 R q t q C = 0 q=q(1-e -t/(rc) ) Q At τ = RC the charge is (1-e -1 )Q=0.632Q
Plumber s RC Analogy Valve Constriction P 1 P 2 Pump Rubber Diaphragm The plumber s analogy of an RC circuit is a pump (=battery) pumping water in a closed loop of pipe that includes a valve (=switch), a constriction (=resistor), and a rubber diaphragm. When the valve starts the flow, the diaphragm stretches until the pressure difference across the pump (P 1 -P 3 ) equals that across the diaphragm (P 2 -P 3 ). P 3 Pump = Battery Valve = Switch Constriction = Resistor Capacitor= Rubber Diaphragm Pressure = Potential Water Flow = Current
RC Circuits: Charging The switches are originally open and the capacitor is uncharged. Then switch S 1 is closed. KVL: Just after S 1 is closed: Capacitor is uncharged (no time has passed so charge hasn t changed yet) ε + - I + - + - R I C S 2 Long time after: Capacitor is fully charged q RC S 1 2RC Intermediate (more complex) q(t) = q (1-e -t/rc ) I(t) = I 0 e -t/rc q 0 t
Summary Discharging Charge: q = Q e -t/(rc) Voltage: V = V 0 e -t/(rc) Current: I = I 0 e -t/(rc) I = I 0 e -t/τ Charging Charge: q=q (1 - e -t/(rc) ) Voltage: V=V 0 (1 - e -t/(rc) ) Current: I = I 0 e -t/(rc) ) I = I 0 e -t/τ Time Constant τ = RC Large τ means long time to charge/discharge Short term: Charge doesn t change (often zero or max) Long term: Current through capacitor is zero. Note: Formula for I is the same for both discharging and charging!
Practice! + R - Calculate current immediately after switch is closed: + ε - I 0 R - q 0 /C = 0 + ε - I 0 R - 0 = 0 I 0 = ε /R Calculate current after switch has been closed for 0.5 seconds: 0 t I = I e RC = ε R 0.5 e RC 20 = 10 e 0.5 10 0.03 Calculate current after switch has been closed for a long time: After a long time current through capacitor is zero! Calculate charge on capacitor after switch has been closed for a long time: - ε + IR + q /C = 0 - ε + 0 + q /C = 0 q = εc I + - 0.5 20 10 0.03 10 e = = 0.38 Amps E S 1 C R=10Ω C=30 mf E =20 Volts + -
Charging: Intermediate Times Calculate the charge on the capacitor 3 10-3 seconds after switch 1 is closed. q(t) = q (1 - e -t/rc ) = q (1 - e -3 10-3 /(20 100 10-6) ) ) = q (0.78) Recall q = C = (50)(100x10-6 ) (0.78) ε + - I b R = 20 Ω = 50 Volts C = 100µF + 2R - + C - R S 2 = 3.9 x10-3 Coulombs S 1
Time Constant Question Each circuit has a 1 F capacitor charged to 100 Volts. When the switch is closed: Which system will be brightest? Which lights will stay on longest? Which lights consumes more energy? 2 (I=2V/R) 1 Same U=1/2 CV 2 1 τ = 2RC 2 τ = RC/2
Question 1 The time constant for the discharge of the capacitor is: (a) 5 s; (b) 4 s; (c) 2 s; (d) 1 s; (e) the capacitor does not discharge because the resistors cancel.
Household Circuits In a typical installation, the utility company distributes electric power to individual houses with a pair of wires, or power lines. The voltage between the two wires is 120 V. Connected in parallel Fig. 18.19, p.570 Slide 32
Circuit breaker In modern homes, circuit breakers are used in place of fuses. When the current exceeds some value, the circuit breaker acts as a switch and opens the circuit. Fig. 18.20, p.571 Slide 33
Circuit breaker, cont. Current passes through a bimetallic strip, the top of which bends to the left when excessive current heats it and settles in a groove in the spring-loaded metal bar. When this occurs, the bar drops enough to open the circuit at the contact point. In contrast to fuses, circuit breakers recover and are reusable.
Heavy-duty appliances Electric ranges and cloth dryers require 240 V to operate. The power company supplies this voltage by providing +120 V and 120 V above and below ground, respectively. Fig. 18.21a, p.571 Slide 34 Fig. 18.21b, p.571 Slide 35
Electrical Safety Electric shock can result in fatal burns Electric shock can cause the muscles of vital organs (such as the heart) to malfunction The degree of damage depends on the magnitude of the current the length of time it acts the part of the body through which it passes
Electrical Safety, cont.
Electrical Safety, cont. Avoid the touch of high potential as an appliance. Possibility of accidents are reduced by dedicated grounding.
Electrical Safety, cont. Put fuses always in the hot side (a) A third wire is connected from an appliance tool to ground. (b) If a loose internal wire comes in contact with the grounded casing, the shortening to ground blows the fuse.
Grounding and GFI Modern power wiring includes a ground line, the round 3 rd wire of an electrical plug. The ground point defines a point of zero potential, which is normally connected directly to the Earth (V earth =0). The operation of any circuit depends only on potential differences, so it should not be affected by the presence or absence of a ground connection. Because the ground connection is connected an only one point, no current should flow through the ground connection. However, if some other part of a circuit is accidentally grounded, current is likely to flow through the ground line. GFI (ground fault interruption) circuits, widely used, e.g., in bathroom wiring, detect current flow in the ground line and interrupt power automatically when it occurs. This has prevented many accidental electrocutions.
Example: A Grounded Circuit The circuit shown is grounded at the junction between the two resistors rather than at the bottom. Find the potential at each corner of the circuit. I = E 10 V R = 8 Ω + 12 Ω = 0.5 A V 1 = (8 Ω )(0.5 A) = 4 V V 2 = (12 Ω )(0.5 A) = 6 V
Ground Fault Interrupts (GFI) Special power outlets Used in hazardous areas Designed to protect people from electrical shock Senses currents (of about 5 ma or greater) leaking to ground Shuts off the current when above this level
Electrical Signals in Neurons The most remarkable use of electrical phenomena in living organisms is found in the nervous system. Specialized cells in the body called neurons form a complex network that receives, processes, and transmits information from one part of the body to another.
Neuron, cont. The nervous system is very complex and consist of 10 10 interconnected neurons. Neurons are the basic units of the nervous system. Over the past 45 years, the methods of signal propagation through the nervous system has been established. The messages are voltage pulses called action potentials transmitted by neurons.
Neuron, cont. Three classes of neurons Sensory neurons Receive stimuli from sensory organs that monitor the external and internal environment of the body Motor neurons Carry messages that control the muscle cells Interneurons Transmit information from one neuron to another
Diagram of a Neuron Each neuron consist of a cell body with input ends called dendrites and a long tail called axon, which transmits the signal away from the cell.
Simple neural circuit The figure shows a simple sensory-motor neuron circuit. A stimulus from a muscle produces nerve impulses that travel to the spine. Fig. 18.24, p.574 Slide 40
Investigation of the electrical and chemical properties of the axon Much information about the chemistry and the electrical properties of the axon is obtained by inserting small needle-like probes into the axon. The figure shows the experimental setup. With such probes it is possible to cause a current through the axon and to measure the resulting action potential. Fig. 18.26, p.574 Slide 41
End of Lecture 14 End of Lecture 14 Before the next lecture, read and study Sections 20.12-3 and 20.14.