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Chapter 25

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Chapter 25

Electric Circuits

- When the current in a circuit has a constant direction, the current is called direct current
- Most of the circuits analyzed will be assumed to be in steady state, with constant magnitude and direction
- Because the potential difference between the terminals of a battery is constant, the battery produces direct current
- The battery is known as a source of emf

- The source that maintains the current in a closed circuit is called a source of emf
- Any devices that increase the potential energy of charges circulating in circuits are sources of emf (e.g. batteries, generators, etc.)
- The emf (ε) is the work done per unit charge
- Emf of a source is the maximum possible voltage that the source can provide between its terminals
- SI units: Volts

- A circuit is a collection of objects usually containing a source of electrical energy connected to elements that convert electrical energy to other forms
- A circuit diagram – a simplified representation of an actual circuit – is used to show the path of the real circuit
- Circuit symbols are used to represent various elements

- When two or more resistors are connected end-to-end, they are said to be in series
- The current is the same in all resistors because any charge that flows through one resistor flows through the other
- The sum of the potential differences across the resistors is equal to the total potential difference across the combination

- The equivalent resistance has the effect on the circuit as the original combination of resistors (consequence of conservation of energy)
- For more resistors in series:
- The equivalent resistance of a series combination of resistors is greater than any of the individual resistors

- A real battery has some internal resistancer; therefore, the terminal voltage is not equal to the emf
- The terminal voltage: ΔV = Vb – Va
ΔV = ε – Ir

- For the entire circuit (R – load resistance):
ε = ΔV + Ir

= IR + Ir

ε = ΔV + Ir= IR + Ir

- ε is equal to the terminal voltage when the current is zero – open-circuit voltage
I = ε / (R + r)

- The current depends on both the resistance external to the battery and the internal resistance
- When R >> r, r can be ignored
- Power relationship: I ε= I2 R + I2 r
- When R >> r, most of the power delivered by the battery is transferred to the load resistor

- The potential difference across each resistor 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 (conservation of charge)
- The currents are generally not the same

- For more resistors in parallel:
- 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
- The equivalent is always less than the smallest resistor in the group

- Combine all resistors in series
- They carry the same current
- The potential differences across them are not necessarily the same
- The resistors add directly to give the equivalent resistance of the combination:
Req = R1 + R2+ …

- Combine all resistors in parallel
- The potential differences across them are the same
- The currents through them are not necessarily the same
- The equivalent resistance of a parallel combination is found through reciprocal addition:

- A complicated circuit consisting of several resistors and batteries can often be reduced to a simple circuit with only one resistor
- Replace resistors in series or in parallel with a single resistor
- Sketch the new circuit after these changes have been made
- Continue to replace any series or parallel combinations
- Continue until one equivalent resistance is found

- If the current in or the potential difference across a resistor in the complicated circuit is to be identified, start with the final circuit and gradually work back through the circuits (use formula ΔV = I R and the procedures describe above)

What resistance should you place in parallel with a 56-kΩ resistor to make an equivalent resistance of 45 kΩ?

Gustav Kirchhoff

1824 – 1887

- 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(Laws) can be used instead:
- 1) Junction (Node) Rule
- 2) Loop Rule

- Junction (Node) Rule (A statement of Conservation of Charge): The sum of the currents entering any junction must equal the sum of the currents leaving that junction
- Loop Rule (A statement of Conservation of Energy): The sum of the potential differences across all the elements around any closed circuit loop must be zero

I1 = I2 + I3

- 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 transversing the loop
- Record voltage drops and rises as they occur
- If a resistor is transversed in the direction of the current, the potential across the resistor is – IR
- If a resistor is transversed in the direction opposite of the current, the potential across the resistor is +IR

- If a source of emf is transversed in the direction of the emf (from – to +), the change in the electric potential is +ε
- If a source of emf is transversed in the direction opposite of the emf (from + to -), the change in the electric potential is – ε

- 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
- The number of times the junction rule can be used is one fewer than the number of junction points in the circuit
- 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

- 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
- Check your answers

The figure shows a portion of a circuit used to model muscle cells and neurons. All resistors have the same value R = 1.5 MΩ, and the emfs are ε1 = 75 mV, ε2 = 45 mV and ε3 = 20 mV. Find the current through ε3, including its direction.

- If a direct current circuit contains capacitors and resistors, the current will vary with time
- At the instant the switch is closed, the charge on the capacitor is zero
- When the circuit is completed, the capacitor starts to charge, and the potential difference across the capacitor increases, until the charge reaches its maximum (Q = Cε)

- Once the capacitor is fully charged (maximum charge is reached), the current in the circuit is zero, and the potential difference across the capacitor matches that supplied by the battery

- The charge on the capacitor varies with time
q = Q (1 – e -t/RC )

- The time constant, = RC, represents the time required for the charge to increase from zero to 63.2% of its maximum

- In a circuit with a large (small) time constant, the capacitor charges very slowly (quickly)
- After t = 10, the capacitor is over 99.99% charged

- When a charged capacitor is placed in the circuit, it can be discharged

- The charge on the capacitor varies with time
q = Qe -t/RC

- The charge decreases exponentially
- At t = = RC, the charge decreases to 0.368 Qmax; i.e., in one time constant, the capacitor loses 63.2% of its initial charge

An uncharged 10-µF capacitor and a 470-kΩ resistor are in series, and 250 V is applied across the combination. How long does it take the capacitor voltage to reach 200 V?

- An ammeter is used to measure current in line with the bulb – all the charge passing through the bulb also must pass through the meter
- A voltmeter is used to measure voltage (potential difference) – connects to the two ends of the bulb

- 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

- 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 just a few seconds, can be fatal

Answers to Even Numbered Problems

Chapter 25:

Problem 18

200 kJ

Answers to Even Numbered Problems

Chapter 25:

Problem 26

2 A

Answers to Even Numbered Problems

Chapter 25:

Problem 30

1 kW

Answers to Even Numbered Problems

Chapter 25:

Problem 58

7.5 mJ