Chapter 28: DC and RC Circuits Kirchhoff s Rules

Chapter 28: DC and RC Circuits Kirchhoff s Rules

Series Circuits The current is the same in each device. The equivalent resistance of the circuit is the sum of the individual resistances. Parallel Circuits The voltage of each device is the full voltage of the EMF source (the battery) The total current is divided between each path: 1 1 1 = + R R R total 1 2 R = R + R total 1 2

Circuits Problem:Bulbs in Series vs Parallel A circuit contains a 48-V battery and two 240Ω light bulbs. In which circuit does each bulb burn brighter? RULE: THE MORE POWER DISSIPATED IN A BULB, THE BRIGHTER IT IS. P= IV Parallel Bulbs Burn Brighter!

Circuits Problem:Bulbs in Series vs Parallel If a bulb burns out - what happens to the other bulb in each circuit? Does it go out? Is it brighter? Dimmer? Or? In the series circuit, the burned out bulb will short the circuit and the other bulb will go out. In the parallel circuit the other bulb will have the same brightness.

Circuits Problem:3 Bulbs in Parallel If one more bulb is added to each circuit (3 bulbs total), how does the brightness of the bulbs change? Or not? In the parallel circuit, the bulbs DO NOT DIM. WHY? In parallel, each of the three equal bulbs gets the full voltage of the battery source. Is this getting something for nothing? NO! Parallel circuits drain the battery faster!

Circuits Problem:3 Bulbs in Series If one more bulb is added to each circuit (3 bulbs total), how does the brightness of the bulbs change? Or not? In the series circuit, the bulbs DIM. WHY? V = V1+ V2 + V3 In series, each of the three equal bulbs gets one third of the Voltage (V/3) that a single bulb would get. P 2 2 V /3 1 V P = = = = R R 9 R 9 ( V ) 2 singlebulb Note: P=VI but I is due to the equivalent Resistance: I = V/R s =V/3R So the Current through each is 1/3 the current through a single bulb and P=VI=V/3 x I/3 = VI/9 = P/9. The bulbs burn 1/9 as bright!

Find Everything V = 15.0V, R = 10.0 Ω, R = 20.0 Ω, R = 30.0 Ω, 1 2 2

YOU DO IT. For the circuit shown, find the equivalent resistance of the circuit, the total current drawn by the battery, and the current through and the potential difference across each resistor. Place your results in a table for ease of reading.

Which way does the current flow?

Single Circuit: Multiple Batts What is the direction of current? Book: Chooses Wrong Current Directions to show you that the rule still works, you get negative currents. But try to pick the correct direction! Book I The 12 V WINS! When polarities of the batteries are opposed, one gets CHARGED.

What is the current in the resistors?

Kirchhoff s Rules When resistors are connected so that the circuits formed cannot be reduced to a single equivalent resistor you can use two rules, called Kirchhoff s rules, to solve the problem by generating systems of equations to find the unknowns!!

Gustav Kirchhoff 1824 1887 German physicist Worked with Robert Bunsen They Invented the spectroscope and founded the science of spectroscopy Discovered the elements cesium and rubidium Invented astronomical spectroscopy

Kirchhoff s Rules for Spectra: 1859 German physicist who developed the spectroscope and the science of emission spectroscopy with Bunsen. Kirkoff Bunsen * Rule 1 : A hot and opaque solid, liquid or highly compressed gas emits a continuous spectrum. * Rule 2 : A hot, transparent gas produces an emission spectrum with bright lines. * Rule 3 : If a continuous spectrum passes through a gas at a lower temperature, the transparent cooler gas generates dark absorption lines.

Kirchhoff s Junction Rule Junction Rule The sum of the currents at any junction must equal zero Currents directed into the junction are entered into the -equation as +I and those leaving as -I A statement of Conservation of Charge Mathematically, junction I = 0

More about the Junction Rule I 1 - I 2 - I 3 = 0 Required by Conservation of Charge Diagram (b) shows a mechanical analog

Kirchhoff s Loop Rule Loop Rule The sum of the potential differences across all elements around any closed circuit loop must be zero A statement of Conservation of Energy Mathematically, Δ V = closed loop 0

Using the Loop Rule Traveling around the loop from a to b In (a), the resistor is traversed in the direction of the current, the potential across the resistor is IR In (b), the resistor is traversed in the direction opposite of the current, the potential across the resistor is is + IR

Loop Rule In (c), the source of emf is traversed in the direction of the emf (from to +), and the change in the electric potential is +ε which means you are CHARGING the battery! In (d), the source of emf is traversed in the direction opposite of the emf (from + to -), and the change in the electric potential is ε I Discharging Charging I

Junction Equations I = 0 from Kirchhoff s Rules junction Use the junction rule as often as needed, so long as each time you write an equation, you include in it a current that has not been used in a previous junction rule equation In general, the number of times the junction rule can be used is one fewer than the number of junction points in the circuit

Loop Equations from Kirchhoff s Rules Δ V = closed loop 0 The loop rule can be used as often as needed so long as a new circuit element (resistor or battery) or a new current appears in each new equation You need as many independent equations as you have unknowns

Kirchhoff s Rules Equations In order to solve a particular circuit problem, the number of independent equations you need to obtain from the two rules equals the number of unknown currents Any capacitor acts as an open branch in a circuit The current in the branch containing the capacitor is zero under steady-state conditions

Kirchhoff s Rules Junction Rule: junction Loop Rule: I = 0 Δ V = closed loop 0 First use junction rule and assign values to the current - guess the directions! Then use the loop rule CONSISTENTLY (Clockwise) on each loop. Any capacitor acts as an open branch in a circuit once under steady state conditions

Multiple Loop Circuit Find the Currents! Book: Chooses Wrong Current Directions to show you that the rule still works, you get negative currents. But try to pick the correct direction! I 1 I 2 I 1 I 3 I 3

Kirchhoff Problem 22. Taking R = 1.00 km and EMF = 250V, determine the direction and magnitude of the current in the horizontal wire between a and e.

Many-Loop Many Equations! You can solve systems of equations with simple algebra or by using linear algebra (Cramer s Rule) but I can t teach you that! I ve put a document on the website if you want to teach yourself. BUT YOU DON T NEED TO! Algebra will work and I won t put a circuit with more than two loops on the exam!

#26 Hint: Req=ΔVab/I Let Δ Vab =V, Assign currents and solve for I! 26/28 For the network shown, show that the resistance R ab = (27/17) Ω.

Electromotive Force: EMF: ε ε = ΔV Ir The emf ε is equivalent to the open-circuit voltage This is the terminal voltage when no current is in the circuit This is the voltage labeled on the battery The actual potential difference ΔV between the terminals of the battery depends on the current in the circuit and the internal resistance of the battery: ΔV = ε Ir If the internal resistance is zero, the terminal voltage equals the emf : ΔV = ε The external resistor is called the load resistance

Power The total power output of the battery is = IΔ V = Iε This power is delivered to the external resistor (I 2 R) and to the internal resistor (I 2 r) is maximum when R = r!! (see nice proof in text) 2 2 = IR+ Ir

What is the emf and internal resistance of the battery?

RC Circuits: Charging The capacitor continues to charge until it reaches its maximum charge (Q = Cε). Once the capacitor is fully charged, the current in the circuit is zero and the The potential difference across the capacitor matches that supplied by the battery.

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=31

RC Circuit http://www.intmath.com/differential-equations/6_rc-circuits.php

RC Circuit: Charging The charge on the capacitor varies with time q(t) = Cε(1 e -t/rc ) = Q(1 e -t/rc ) τ is the time constant τ = RC The current can be found I( t) = ε R e trc The time constant τ has units of time represents the time required for the charge to increase from zero to 63.2% of its maximum The energy stored in the charged capacitor is ½ Qε = ½ Cε 2

RC Circuit: Charging q(t) = Q(1 e -t/rc ) I( t) = ε R e trc

Discharging a Capacitor in an RC Circuit When a charged capacitor is placed in the circuit, it can be discharged q = Qe -t/rc The charge decreases exponentiallyat t = τ = RC, the charge decreases to 0.368 Q max In other words, in one time constant, the capacitor loses 63.2% of its initial charge The current is: I () t dq Q = = dt RC e trc

HO RC Problem

Another RC Problem The switch in the figure has been in position a for a long time. It is changed to position b at t = 0s. What are the charge on the capacitor and the current I through the resistor a)immediately after the switch is closed? b)what is the time constant τ? c)at t = 50 μs? d)at t = 200 μs?

Electric Shock What causes electric Shock in the human body, Voltage or Current? Electric Shock occurs when current is produced in the body, which is caused by an impressed voltage. Voltage is the CAUSE Current does the DAMAGE

Electric Shock Current (A) Effect 0.001 Can be felt 0.005 Painful 0.010 Causes involuntary muscle spasms 0.015 Causes loss of muscle control 0.070 If through heart, serious! If current lasts for 1 s - FATAL! Dry Skin Body Resistance: 500,000 Ω Wet Skin Body Resistance: 1000 Ω

Electrical equipment manufacturers use electrical cords that have a third wire, called a ground This safety ground normally carries no current and is both grounded and connected to the appliance Ground Wire

Ground Wire, cont If the live wire is accidentally shorted to the casing, most of the current takes the lowresistance path through the appliance to the ground If it was not properly grounded, anyone in contact with the appliance could be shocked because the body produces a low-resistance path to ground

Ground-Fault Interrupters (GFI) Special power outlets Used in hazardous areas Designed to protect people from electrical shock Senses currents (< 5 ma) leaking to ground Quickly shuts off the current when above this level

Is AC Deadlier than DC? Low frequency (50-60 Hz) AC currents can be more dangerous than similar levels of DC current since the alternating fluctuations can cause the heart to lose coordination, inducing ventricular fibrillation, which then rapidly leads to death. High voltage DC power can be more dangerous than AC, however, since it tends to cause muscles to lock in position, stopping the victim from releasing the energised conductor once grasped.

Frequency Matters

Electric Shock Therapy ELECTRO CONVULSIVE THERAPY An electric shock is applied to produce a convulsive seizure. The shock is typically between 140-170 volts and lasts between 0.5 and 1 seconds. No explanation of how it works. Used in the treatment of: 1.Chronic endogenous depression 2.Bipolar disorder. 3.Acute mania. 4.Certain types of schizophrenia In the U.S. 33,000-50,000 people receive ECT each year