Chapter 23 Electric Circuits Neurons connected together to form the electrical circuitry in the brain

Contents Circuit Elements and Schematic Diagrams Series Circuits Parallel Circuits Supplemental Topics: Capacitor Circuits RC Circuits Electricity in the Nervous System

Circuit Elements and Schematic Diagrams

Some common circuit elements and their schematic symbols

Which diagrams represent the same circuit?

Series Circuits

More than 1 Resistor: “Series Circuit” How much current flows thru each resistor? Is the current thru each resistor different? If we recall the water-pipe analogy, the current thru the circuit must be continuous: The current is the same everywhere.

Series Circuit (continued) How much current flows? Equivalent circuit: Add resistors in series:

Series Circuit (continued) What happens when we apply Ohm’s Law to each resistor in the series circuit? The sum of resistor voltages around a series circuit equals the battery voltage. Conservation of energy: The energy added by the battery = energy lost thru resistors (heat dissipation) This is known as Kirchoff’s Loop Law

4 Rules for Solving Series Circuits Same current I flows thru entire circuit Total or “effective” R is the sum of all R’s Use Ohm’s law (V=IR, I =V/R, R=V/I) for each resistor as needed. Sum of resistor voltages = Battery voltage

Example Problems 3 amperes flows thru the 4-ohm resistor in the circuit shown below. The voltage across R 2 is measured to be 18 volts. 1.How much current flows thru R 2 ? 3 A 2.What is the value of R 2 ? R 2 = V 2 /I 2 = 18/3 = 6 ohms 3.What is the voltage across the 4 ohm resistor? V = IR = 12 V 4.What is the total circuit resistance? = 10 ohms 5.Determine the battery voltage = 30 V or: V = I R tot = (3 A)(10 Ω) = 30 V

Parallel Circuits

Consider a simple circuit with a 12-volt battery and a 4 Ω resistor. By Ohm’s law: I = V/R = 12/4 = 3 amperes Now suppose we add another 4 Ω resistor parallel to the first. This is called a parallel circuit. How much current flows thru the parallel circuit? Less, More, or the same ??

Resistors in Parallel Adding resistance in parallel results in more current because it creates another path for current to flow thru: The parallel circuit has less resistance than if just one resistor were present!

Parallel Circuit: Supermarket Analogy The cash register is resistance to the flow (current) of customers: Adding another cash register in parallel actually increases the flow of customers by reducing the overall resistance.

Analysis of Parallel Circuits The same voltage appears across each resistor in a parallel circuit, because they are all connected directly to the battery: The current thru the battery is the sum of currents flowing thru the parallel resistors. This is known as Kirchoff’s Junction Law

Analysis of Parallel Circuits (2) Equivalent circuit: V = I R R = V/I = 12/11=1.1 Ω

4 Rules for Solving Parallel Circuits The battery voltage appears across each resistor Total current thru battery = sum of parallel resistor currents: I total = I 1 + I 2 + … Total or “effective” R is found by Ohm’s law: R total = V battery /I total or use the “Handy Rule” shown soon. Use Ohm’s law (V=IR, I =V/R, R=V/I) for each resistor as needed.

Analysis of Parallel Circuits (3) What is the value of resistor R 1 in the diagram? Voltage across R1: 12 V Voltage across R2: 12 V Current thru R2: I 2 = V 2 /R 2 = 12/6 = 2 A Current thru R1: I 1 + I 2 = 6 I 1 = 4 A Resistance of R1: R 1 = V 1 /I 1 = 12/4 = 3 Ω Ohm’s Law V = I R

A Handy Rule for Parallel Resistors

Combination Series- Parallel Circuits What is the equivalent resistance of this resistor network? SOLUTION:

Another tricky example Find the current that flows thru each resistor:

Solution

Measuring Voltage, Current, and Resistance

Measuring Voltage, Current, and Resistance with a Multimeter Multimeter: Handy and practical measurement tool that can be used to measure voltages, currents, and resistances in an electric circuit.

Measuring Circuit Voltages: Voltmeter 1.Set the multimeter selector to “DC Voltage” 2.Touch the probes to the region of the circuit you wish to measure the potential difference (voltage). Example: Measure the voltage across the 24-ohm resistor.

Measuring Circuit Current: Ammeter 1.Set the multimeter selector to “DC Current” 2.Break the circuit at the point you wish to measure current. Insert the multimeter probes so that the circuit current flows thru the multimeter. (The multimeter has no resistance so it does not effect the circuit!). Example: Measure the current in the circuit shown.

Measuring Circuit Resistances 1.Set the multimeter selector to “Resistance” 2.Remove the resistance (lightbulb, resistor, etc) you wish to measure from the circuit. 3.Connect the multimeter probes to the resistor. Example: Measure each of the resistors in the circuit disconnected

Supplemental Topic: Capacitor Circuits

Capacitors in Parallel and Series Note the formulas for series and parallel capacitors are opposite of those for series and parallel resistors!

Example: Simplify a capacitor circuit

Supplemental Topic: RC Circuits

The RC Circuit Circuits containing both resistors (R) and capacitors (C) Discharging a capacitor thru a resistor: I = 0

RC Circuit in Discharge

The RC Time Constant Exponential decay is characterized by a time constant τ = RC: The time for the exponential function to decay to 37% of its initial value. Larger values of R or C increase the amount of time it takes to discharge the capacitor: Larger C means more charge to move. Larger R means less discharge current

Charging a Capacitor in an RC Circuit When the switch is first closed, there is no potential difference across the capacitor (ΔV=0) because there is no charge stored yet. Thus the capacitor behaves momentarily like a short circuit: VbVb VbVb After the switch is closed, the capacitor charges. Advanced theory tells us the current falls exponentially as the capacitor charges up: The capacitor voltage ΔVc is found from Ohm’s Law: Vb- ΔVc = I R. Solving for ΔVc and substituting the results shown above we get:

Supplemental Topic: Electricity in the Nervous System

Electricity in the Nervous System Neurons connected together to form the complex electrical circuitry in the brain of a mouse. We can model the electrical connections as an RC circuit. Human have ~ 300 Billion neurons

A Simple Model of a Nerve Cell (Neuron) Scientific Model: A simplified way to represent a complex system. The purpose of the model is to describe specific features that are observed in the actual system. The cell membrane has “pumps” and “channels” that move electrical charge (Na and K ions) in and out of the cell

Separation of Charge Produces Potential Differences The ion pump acts like a battery, moving and separating charged ions, producing a voltage between the inside and outside of the cell. “resting potential”

Resistance of the Cell Membrane Very high resistance  Good insulator!

Capacitance of the Cell Membrane

The Cell Membrane as an RC Circuit This time constant (3 ms) refers to the nerve cell’s ability to store charge (capacitance) and discharge thru the resistance (cell membrane) if left alone. We will next see that the neuron can react much faster than this time constant when stimulated. Time constant of the circuit:

The Action Potential: Neuron“Firing” The neuron normally sits quietly at its “resting potential” of 70 mV. Neurotransmitters released at synapses (junctions between neurons) cause a “stimulus” to the neuron, causing it to “fire” like an electrical gun. Firing neuron sends stimulus to next neuron, etc.

The Action Potential: Neuron Firing Depolarization: Stimulus causes Sodium channels to open. Positive charge entering cell raises potential Repolarization: Potassium channels open. Positive K ions leave cell, lowering potential. Resting Potential Ion current thru cell membrane (RC circuit)

Motor Neuron Signals from brain to muscles are propagated along motor neurons thru the axons (chain up to 1 meter long)

How Nerve Impulses Travel thru the body

Nerve Impulse moving along the nodes of an axon

Modeling the Axon Nodes as RC Circuits

Speed of the Nervous System: Reaction Time The time constant τ is an estimate of the time it takes for the nerve signal to jump from one axon node to the next. The speed of the nerve impulse is then: Example: How long does it take to react when you touch a hot surface on a stove? The distance between your brain and your hand is about 1 meter. The nerve impulse must travel 1 meter to your brain, then 1 meter back to your hand: