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W10D2DC Circuits

Today’s Reading Assignment W10D2 DC Circuits & Kirchhoff’s Loop Rules Course Notes: Sections 7.1-7.5

Announcements

PS 8 due Week 11 Tuesday at 9 pm in boxes outside 32-082 or 26-152

Next Reading Assignment W10D3 PS07: PhET: Building a Circuit 7.1-7.5, 7.10

Exam 3 Thursday April 18 7:30 pm –9:30 pm

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Outline

DC Circuits

Kirchoff’s Laws

Electrical Power

Measuring Voltage and Current

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DC Circuits

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Examples of Circuits

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Symbols for Circuit Elements

Battery

Resistor

Capacitor

Switch

Equipotential

Junction

Branch

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Electromotive Force (EMF)

The work per unit charge around a closed path done is called electromotive force EMF. This is a bad name because it is not a force. Let denote the force per unit charge, then the EMF is

If a conducting closed path is present then

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Ideal Battery

1) Inside Battery: chemical force , non-zero inside battery (zero outside), moves charges through a region in which static electric field opposes motion

2) Ideal battery: net force on charges is zero

3) Potential difference between its terminals

4) Extend path through external circuit where

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Concept Question

The electric field inside the battery points to

1. The left

2. The Right

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Concept Question Answer

The electric field inside the battery points to

1. The left

Goes from positive to negative charge

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Sign Conventions - BatteryMoving from the negative to positive terminal of a battery increases your potential

Moving from the positive to negative terminal of a battery decreases your potential

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Concept Question

The electric field inside the resistor points to

1. The left

2. The Right

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Concept Question Answer

The electric field inside the resistor points to

2. The right

Electric field pushes charge in the same direction, so current in direction of E

This is cause and effect!

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Sign Conventions - ResistorMoving across a resistor in the direction of current decreases your potential

Direction of current

Is the same as direction of electric field which points to lower electric potential

Moving across a resistor in the direction of current decreases your potential

Internal Resistance

Real batteries have an internal resistance, r, which is small but non-zero

Terminal voltage:

(Even if you short the leads you don’t get infinite current)

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Series vs. Parallel

Series Parallel

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Resistors In SeriesThe same current I must flow through both resistors

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Current Conservation

Sum of currents entering any junction in a circuit must equal sum of currents leaving that junction.

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Resistors In ParallelVoltage drop across the resistors must be the same

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Concept Q: Resistors In Parallel

Suppose that . Then the equivalent resistance

1.

2.

3.

4.

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Concept Q: Resistors In Parallel

Answer 1. When

Then the equivalent resistance is the smaller resistance. This is an important circuit design principle.

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Group Problem: Four Resistors

Four resistors are connected to a battery

as shown in the figure. The current in the battery is I,

the battery emf is ε, and the resistor values are R1 = R, R2 = 2R, R3 = 4R, R4 = 3R.

Determine the current in

each resistor in terms of I.

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Measuring Voltage & Current

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Measuring Potential Difference

A voltmeter must be hooked in parallel across the element you want to measure the potential difference across

Voltmeters have a very large resistance, so that they don’t affect the circuit too much

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Measuring Current

An ammeter must be hooked in series with the element you want to measure the current through

Ammeters have a very low resistance, so that they don’t affect the circuit too much

P18-26

Concept Question: Measuring Current

If R1 > R2, compare the currents measured by the three ammeters:

1. I1 > I2 > I3

2. I2 > I1 > I3

3. I3 > I1 > I2

4. I3 > I2 > I1

5. I3 > I1 = I2

6. None of the above7. Not enough information is given.

P18-27

Concept Question Answer: Measuring Current

Answer: 4. I3 > I2 > I1

The total current must add to the two individual currents, so I3 must be largest. Most current prefers to go through the smaller resistor so I2 > I1 .

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Measuring Resistance

An ohmmeter must be hooked in parallel across the element you want to measure the resistance of

Here we are measuring R1

Ohmmeters apply a voltage and measure the current that flows. They typically won’t work if the resistor is powered (connected to a battery)

P18-29

Concept Question: Bulbs & Batteries

An ideal battery is hooked to a light bulb with wires. A second identical light bulb is connected in parallel to the first light bulb. After the second light bulb is connected, the current from the battery compared to when only one bulb was connected.

1. Is Higher2. Is Lower3. Is The Same4. Don’t know

P18-30

Concept Question Answer: Bulbs & Batteries

There are several ways to see this:

(A) The equivalent resistance of the two light bulbs in parallel is half that of one of the bulbs, and since the resistance is lower the current is higher, for a given voltage.

(B) The battery must keep two resistances at the same potential I doubles.

Answer: 1. More current flows from the battery

P18-31

Concept Question: Bulbs & Batteries

An ideal battery is hooked to a light bulb with wires. A second identical light bulb is connected in series with the first light bulb. After the second light bulb is connected, the current from the battery compared to when only one bulb was connected. 1. Is Higher

2. Is Lower

3. Is The Same

P18-32

Concept Question Answer: Bulbs & Batteries

The equivalent resistance of the two light bulbs in series is twice that of one of the bulbs, and since the resistance is higher the current is lower, for the given voltage.

Answer: 2. Less current flows from the battery

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Electrical Power

Power is change in energy per unit time

So power to move current through circuit elements:

VIP

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Power - Battery

I

IVIP supplied

Moving from the negative to positive terminal of a battery increases your potential. If current flows in that direction the battery supplies power

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Power – Resistor (Joule Heating)

Moving across a resistor in the direction of current decreases your potential. Resistors always dissipate power

P18-36

Concept Question: PowerAn ideal battery is hooked to a light bulb with wires. A second identical light bulb is connected in parallel to the first light bulb. After the second light bulb is connected, the power output from the battery (compared to when only one bulb was connected)

1. Is four times higher

2. Is twice as high

3. Is the same

4. Is half as much

5. Is ¼ as much

P18-37

Concept Question Answer: Power

Answer: 2. Is twice as high

The current from the battery must double (it must raise two light bulbs to the same voltage difference) and

P18-38

Concept Question: Power

An ideal battery is hooked to a light bulb with wires. A second identical light bulb is connected in series with the first light bulb. After the second light bulb is connected, the light (power) from the first bulb (compared to when only one bulb was connected)

1. Is four times higher

2. Is twice as high

3. Is the same

4. Is half as much

5. Is ¼ as much

P18-39

Concept Question Answer: Power

Answer: 5. Is 1/4 as bright

R doubles current is cut in half. So power delivered by the battery is half what it was. But that power is further divided between two bulbs now.

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Kirchhoff’s Loop Rules

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Recall: Current Conservation

Sum of currents entering any junction in a circuit must equal sum of currents leaving that junction.

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Sum of Potential Differences Around a Closed Path

Sum of potential differences across all elements around any closed circuit loop must be zero.

0i statici Closed

Path

V d E s

NO TIME DEPENDENCE! 0Bd

dt

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Sum of Potential Differences Around a Closed Path

Sum of potential differences across all elements around any closed circuit loop must be zero.

0i statici Closed

Path

V d E s

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Demonstration:Five Different Types of Resistance

F13

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F 13&show=0

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Steps of Solving Circuit Problem

1. Straighten out circuit (make squares)

2. Simplify resistors in series/parallel

3. Assign current loops (arbitrary)

4. Write loop equations (1 per loop)

5. Solve

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Worked Example: Circuit

What is current through each branch of the circuit shown in the figure below?

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Worked Example: Simple Circuit

Upper Loop: Start at a and circulate counterclockwise. Then

Current conservation at a:

Lower Loop: Start at a and circulate clockwise. Then

Solve for I2: and

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Demonstration:Wheatstone Bridge F14

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F 14&show=0

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Group Problem: Wheatstone Bridge

A circuit consisting of two resistors with R1=6.0 ohms and R2=1.5 ohms, a variable resistor, with resistance Rvar, a resistor of unknown value Ru, and 9.0 volt battery, are connected as shown in the figure. When Rvar is adjusted to 12 ohms, there is zero current through the ammeter. What is the unknown resistance Ru?