Series and Parallel Circuits Direct Current Circuits.

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Series and Parallel Circuits Direct Current Circuits

Transcript of Series and Parallel Circuits Direct Current Circuits.

Page 1: Series and Parallel Circuits Direct Current Circuits.

Series and Parallel Circuits

Direct Current Circuits

Page 2: Series and Parallel Circuits Direct Current Circuits.

Series circuits• A series circuit is formed

in a circuit like this one, where current flows through one path that travels through one device after another (the two light bulbs).

• If you measure the current at any point in this circuit, it will be the same.

Page 3: Series and Parallel Circuits Direct Current Circuits.

Resistors in Series

When two or more resistors are connected end-to-end, they are said to be in series.

The current is the same in resistors in series because any charge that flows through one resistor flows through the other.

The sum of the potential differences (voltage drops) across the resistors is equal to the total potential difference across the combination (the battery or power supply voltage).

Page 4: Series and Parallel Circuits Direct Current Circuits.

Resistors in Series

Potentials add ΔV = IR1 + IR2

= I (R1+R2) Consequence of

Conservation of Energy

The equivalent resistance R1+R2 has the same effect on the circuit as the original combination of resistors

Page 5: Series and Parallel Circuits Direct Current Circuits.

Equivalent Resistance – Series

Req = R1 + R2 + R3 + … The equivalent resistance of a series

combination of resistors is the algebraic sum of the individual resistances and is always greater than any of the individual resistances.

Note that the equivalent resistance is always LARGER than any of the individual resistances when in series.

Page 6: Series and Parallel Circuits Direct Current Circuits.

Equivalent Resistance – SeriesAn Example

Four resistors are replaced with their equivalent resistance.

The current

Page 7: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 1

When a piece of wire is used to connect points b and c in this figure, the brightness of bulb R1 (a) increases, (b) decreases, or (c) stays the same. The brightness of bulb R2 (a) increases, (b) decreases, or(c) stays the same.

Page 8: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 1 ANSWERR1 becomes brighter. Connecting a wire from b to c provides a nearly zero resistance path from b to c and decreases the total resistance of the circuit from R1 + R2 to just R1.

Ignoring internal resistance, the potential difference maintained by the battery is unchanged while the resistance of the circuit has decreased. The current passing through bulb R1 increases, causing this bulb to glow brighter.

Bulb R2 goes out because essentially all of the current now passes through the wire connecting b and c and bypasses the filament of Bulb R2.

Page 9: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 2With the switch in this circuit (figure a) closed, no current exists in R2 because the current has an alternate zero-resistance path through the switch. Current does exist in R1 and this current is measured with the ammeter at the right side of the circuit. If the switch is opened (figure b), current exists in R2. After the switch is opened, the reading on the ammeter (a) increases, (b) decreases, (c) does not change.

Page 10: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 2 ANSWER

(b). When the switch is opened, resistors R1 and R2 are in series, so that the total circuit resistance is larger than when the switch was closed. As a result, the current decreases.

Page 11: Series and Parallel Circuits Direct Current Circuits.

Voltage Divider CircuitFor this circuit, I = V/Req soI = V / (RA + RB) The voltage drop VB, across resistor RB is given by

VB = IRB

or

Page 12: Series and Parallel Circuits Direct Current Circuits.

Resistors in Parallel

For resistors in parallel, the potential difference across each resistor (from a to b) is the same because each is connected directly across the battery terminals.

The current, I, that enters a point must be equal to the total current leaving that point. I = I1 + I2 The currents I1 and I2 are generally not the

same. This is a consequence of conservation of charge.

Page 13: Series and Parallel Circuits Direct Current Circuits.

Equivalent Resistance – Parallel, Examples

Equivalent resistance replaces the two original resistances

Household circuits are wired so the electrical devices are connected in parallel Circuit breakers may be used in series with

other circuit elements for safety purposes

Page 14: Series and Parallel Circuits Direct Current Circuits.

Equivalent Resistance – Parallel Circuits

Equivalent Resistance

The inverse of the equivalent resistance of two or more resistors connected in parallel is the algebraic sum of the inverses of the individual resistance.

321eq R

1

R

1

R

1

R

1

For parallel R’s, the equivalent Req is always less than the smallest resistor in the group.

Page 15: Series and Parallel Circuits Direct Current Circuits.

Problem-Solving Strategy, 1

When two or more unequal resistors are connected in series, they carry the same current, but the potential differences across them are not the same. The resistors add directly to give the

equivalent resistance of the series combination Req = R1 + R2

The equivalent resistance is always larger than any of the individual resistances.

Page 16: Series and Parallel Circuits Direct Current Circuits.

Problem-Solving Strategy, 2 When two or more unequal resistors are

connected in parallel, the potential differences (voltage drop) across them are the same. The currents through them are not the same. The equivalent resistance of a parallel

combination is found through reciprocal addition

The equivalent resistance is always less than the smallest individual resistor in the combination

321eq R

1

R

1

R

1

R

1

Page 17: Series and Parallel Circuits Direct Current Circuits.

Problem-Solving Strategy, 3

A complicated circuit consisting of several resistors and batteries can often be reduced to a simple equivalent circuit with only one resistor Replace any groups of resistors in series or in

parallel using steps 1 or 2. Sketch the new circuit after each change has

been made. Continue to replace any series or parallel

combinations . Continue until one equivalent resistance is

found.

Page 18: Series and Parallel Circuits Direct Current Circuits.

Problem-Solving Strategy, 4

If the current or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit found in step 3 and gradually work back through the circuits Use ΔV = I R and the procedures in

steps 1 and 2

Page 19: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 3With the switch in this circuit (figure a) open, there is no current in R2. There is current in R1 and this current is measured with the ammeter at the right side of the circuit. If the switch is closed (figure b), there is current in R2. When the switch is closed, the reading on the ammeter (a) increases, (b) decreases, or (c) remains the same.

Page 20: Series and Parallel Circuits Direct Current Circuits.

QUICK QUIZ 3 ANSWER(a). When the switch is closed, resistors R1 and R2 are in parallel, so that the total circuit resistance is smaller than when the switch was open. As a result, the total current increases.

Page 21: Series and Parallel Circuits Direct Current Circuits.

Equivalent Resistance – Complex CircuitParallel part:

so Req = 6/3 = 2 Ω

Page 22: Series and Parallel Circuits Direct Current Circuits.

Kirchhoff’s Rules

There are ways in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor

Two rules, called Kirchhoff’s Rules can be used instead

Page 23: Series and Parallel Circuits Direct Current Circuits.

Statement of Kirchhoff’s Rules

Junction Rule The sum of the currents entering any

junction must equal the sum of the currents leaving that junction

A statement of Conservation of Charge

Loop Rule The sum of the potential differences

across all the elements around any closed circuit loop must be zero

A statement of Conservation of Energy

Page 24: Series and Parallel Circuits Direct Current Circuits.

More About the Junction Rule

I1 = I2 + I3 From

Conservation of Charge

Diagram b shows a mechanical analog

Page 25: Series and Parallel Circuits Direct Current Circuits.

Setting Up Kirchhoff’s Rules

Assign symbols and directions to the currents in all branches of the circuit If a direction is chosen incorrectly, the

resulting answer will be negative, but the magnitude will be correct

When applying the loop rule, choose a direction for traversing the loop Record voltage drops and rises as

they occur

Page 26: Series and Parallel Circuits Direct Current Circuits.

More About 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 +IR

Page 27: Series and Parallel Circuits Direct Current Circuits.

Loop Rule, final

In c, the source of emf is traversed in the direction of the emf (from – to +), the change in the electric potential is +ε

In d, the source of emf is traversed in the direction opposite of the emf (from + to -), the change in the electric potential is -ε

Page 28: Series and Parallel Circuits Direct Current Circuits.

Junction Equations from Kirchhoff’s Rules

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

Page 29: Series and Parallel Circuits Direct Current Circuits.

Loop Equations from Kirchhoff’s Rules

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

Page 30: Series and Parallel Circuits Direct Current Circuits.

Problem-Solving Strategy – Kirchhoff’s Rules

Draw the circuit diagram and assign labels and symbols to all known and unknown quantities. Assign directions to the currents.

Apply the junction rule to any junction in the circuit

Apply the loop rule to as many loops as are needed to solve for the unknowns

Solve the equations simultaneously for the unknown quantities. For example 2 equations and 2 unknowns or 3 equations and 3 unknowns

Page 31: Series and Parallel Circuits Direct Current Circuits.

Example 1 by standard method I1

I2

Left loop: 18 - 5I1 - 5I2 - 1.5 I1 = 0Whole loop: 18 - 5I1 - 5 I3 - 5I3 -1.5I1= 0Junction: I1 = I2 + I3

Three equations, three unknowns, solve.You should get I1 = 1.83A, I2 = 1.22A, I3 = 0.61A

I3

I3I1

Page 32: Series and Parallel Circuits Direct Current Circuits.

Example 1- mesh method

I1 I2

Loop 1: 18 - 5I1 - 5(I1-I2) - 1.5 I1 = 0Loop 2: 0 - 5I2 - 5 I2 - 5(I2-I1) = 0By using the mesh method, you have one less equation to solve.Two equations, two unknowns, solve.You should get I1 = 1.83A, I2 = 0.61

Page 33: Series and Parallel Circuits Direct Current Circuits.

Example 2 –stand.method

I1

I2

Full loop: 12 – 1(I1) – 3(I1) – 8(I3) = 0Left loop: 4 – 1(I2) – 5(I2) – 8(I3) = 0Junction: I1 + I2 = I3

3 equations, 3 unknowns, solve for I1, I2, I3

I1 = 1.31A, I2 = -.46A, I3 = 0.85A

Try this one

I1

I3

Page 34: Series and Parallel Circuits Direct Current Circuits.

Example 2mesh method

I1 I2

Loop 1: 4 – 1(I1-I2) – 5(I1-I2) – 8(I1) = 0Loop 2: 12 – 1(I2) – 3(I2) – 5(I2-I1) –

1(I2-I1) – 4 = 02 equations, 2 unknowns, solve for I1, I2

I1 = 0.85, I2 = 1.31, I2 – I1 = 0.46A

Try this one

Page 35: Series and Parallel Circuits Direct Current Circuits.

Example 3 – mesh method

I2

I1

Loop 1: 20 – 30I1 – 5 (I1-I2) – 10 = 0Loop 2: 10 - 5(I2-I1) – 20(I2) = 02 equations, 2 unknowns, solve for I1, I2

I1 = 0.353A , I2 = 0.47A

Try this one

Page 36: Series and Parallel Circuits Direct Current Circuits.

Capacitors

A parallel plate capacitor consists of two metal plates, one carrying charge +q andthe other carrying charge –q.

It is the capacity to store charge that resulted in the name capacitor.

It is common to fill the region between the plates with an electrically insulatingsubstance called a dielectric.

Page 37: Series and Parallel Circuits Direct Current Circuits.

The relation between charge and voltage for a capacitor

• The magnitude of the charge on each plate of the capacitor is directly proportional to the magnitude of the potential difference between the plates.

q = CV

• The capacitance C is the proportionality constant.

• SI Unit of Capacitance: coulomb/volt = farad (F)

• Typical measurements are in microfarads (μF) or picofarads (pF).

Page 38: Series and Parallel Circuits Direct Current Circuits.

Capacitors

THE DIELECTRIC CONSTANT

• The insulating material between the plates is called a dielectric.

• If a dielectric is inserted between the plates of a capacitor, the capacitance can increase markedly.

• Eo and E are , respectively, the magnitudes of the E field between the plates without and with a dielectric.Dielectric constant

𝜿=𝑬𝒐

𝑬

Page 39: Series and Parallel Circuits Direct Current Circuits.

Capacitors

Page 40: Series and Parallel Circuits Direct Current Circuits.

Capacitance for parallel plate capacitors

Parallel plate capacitorfilled with a dielectric

𝐸=𝑉𝑑

=𝐸𝑜

𝜅𝐸𝑜=

𝑞𝜖𝑜𝐴

𝑞=(𝜅𝜖𝑜 𝐴𝑑 )𝑉

ϵo = permittivity of free space = 8.85E-12 C2/N m2

𝐶=𝜅𝜖𝑜 𝐴𝑑

Page 41: Series and Parallel Circuits Direct Current Circuits.

Energy storage in a capacitor

Ad Volume

𝐸𝑛𝑒𝑟𝑔𝑦=12𝑞𝑉=

12

(𝐶𝑉 )𝑉=12𝐶𝑉 2

V =Ed

𝐸𝑛𝑒𝑟𝑔𝑦=12¿

𝐸𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦=𝐸𝑛𝑒𝑟𝑔𝑦𝑣𝑜𝑙𝑢𝑚𝑒

=12𝜅𝜖𝑜𝐸

2

Page 42: Series and Parallel Circuits Direct Current Circuits.

Capacitors in parallel

Parallel capacitors 321 CCCCP

VCCVCVCqqq 212121

Page 43: Series and Parallel Circuits Direct Current Circuits.

Capacitors in series

212121

11

CCq

C

q

C

qVVV

Series capacitors 321

1111

CCCCS

Page 44: Series and Parallel Circuits Direct Current Circuits.

Capacitor charging

Capacitor charging

RCto eqq 1

RC

time constant

Page 45: Series and Parallel Circuits Direct Current Circuits.

Capacitor discharging

Capacitor discharging

RCtoeqq

RC

time constant

Page 46: Series and Parallel Circuits Direct Current Circuits.

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

Page 47: Series and Parallel Circuits Direct Current Circuits.

Effects of Various Currents

5 mA or less can cause a sensation of shock generally little or no damage

10 mA hand muscles contract may be unable to let go a of live wire

100 mA if passes through the body for 1 second or

less, can be fatal

Page 48: Series and Parallel Circuits Direct Current Circuits.

Ground Wire

Electrical equipment manufacturers use electrical cords that have a third wire, called a ground

Prevents shocks

Page 49: Series and Parallel Circuits Direct Current Circuits.

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

Page 50: Series and Parallel Circuits Direct Current Circuits.

Three types of safety devices

There are three types of safety devices that are used in household electric circuits to prevent electric overcurrent conditions: Fuses Circuit Breakers Ground Fault Interrupters