series and parallel circuits direct current circuits
TRANSCRIPT
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.
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).
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
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.
Equivalent Resistance – SeriesAn Example
Four resistors are replaced with their equivalent resistance.
The current
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.
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.
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.
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.
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
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.
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
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.
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.
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
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.
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
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.
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.
Equivalent Resistance – Complex CircuitParallel part:
so Req = 6/3 = 2 Ω
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
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
More About the Junction Rule
I1 = I2 + I3 From
Conservation of Charge
Diagram b shows a mechanical analog
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
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
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
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
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
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
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
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
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
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
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
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.
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).
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
Capacitors
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
Energy storage in a capacitor
Ad Volume
𝐸𝑛𝑒𝑟𝑔𝑦=12𝑞𝑉=
12
(𝐶𝑉 )𝑉=12𝐶𝑉 2
V =Ed
𝐸𝑛𝑒𝑟𝑔𝑦=12¿
𝐸𝑛𝑒𝑟𝑔𝑦 𝑑𝑒𝑛𝑠𝑖𝑡𝑦=𝐸𝑛𝑒𝑟𝑔𝑦𝑣𝑜𝑙𝑢𝑚𝑒
=12𝜅𝜖𝑜𝐸
2
Capacitors in parallel
Parallel capacitors 321 CCCCP
VCCVCVCqqq 212121
Capacitors in series
212121
11
CCq
C
q
C
qVVV
Series capacitors 321
1111
CCCCS
Capacitor charging
Capacitor charging
RCto eqq 1
RC
time constant
Capacitor discharging
Capacitor discharging
RCtoeqq
RC
time constant
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
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
Ground Wire
Electrical equipment manufacturers use electrical cords that have a third wire, called a ground
Prevents shocks
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
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