Chapter 23

Question 1

Who created the two laws?

Answer 1

• Just as our dynamics analysis is based on Newton’s 3 Laws, we have several laws that form the basis of electric circuit analysis. These two laws were developed by German physicist Gustav Kirchhoff in 1845.

Question 2

Series Circuits

Answer 2

The components are connected end to end, so that there is only one path for the current to follow.

Question 3

• Kirchhoff’s Current Law (Junction Rule)

Answer 3

At any junction point in a circuit, the total current entering the junction equals the total current leaving the junction. - This is based on conservation of electric charge. All charge entering a junction must leave the junction. Charges cannot be created or destroyed, and cannot pile up at a point in a circuit.

Question 4

® Kirchhoffs Voltage Law (Loop Rule)

Answer 4

The algebraic sum of the changes in potential around any closed path of a circuit must be zero.

or

Around any complete path through an electric circuit, the sum of the increases in potential is equal to the sum of the decreases in potential. — This is based on conservation of energy. Recall that “potential” is the potential energy per coulomb of charge, so whatever electrical energy the charge has gained as it passes through the source is lost, or more accurately, converted to something else along the way as the charge returns to the source. - Compare this with gravitational potential energy as you go around a loop on a roller coaster. Starting from any point, Ep increases and decreases around the loop, but ends at the same level it started. The total of the changes is zero.

Question 5

Are these laws actually very powerful?

Answer 5

• Although these laws may seem very simple and straightforward, they are actually very powerful and allow us to solve complicated problems such as the following (although we won’t actually do that now).

Question 6

Voltage in series:

Answer 6

The potential decreases some across each resistor, and so the total potential difference produced by the battery (i.e., the emf) is equal to the sum of the potential differences across the resistors.

Vt = V1 + V2 + V3

This agrees with Kirchhoff’s Voltage Law. The potential increases across the battery and then decreases some across each load until it is back to zero at the negative terminal of the battery.

Question 6

Current in Series:

Answer 6

Because every electron goes through every load, the current through each load is equal and is the total circuit current: It = I1 + I2 + I3

How does this relate to Kirchhoff’s Current Law?

→ You could say that there are no junction points, so it doesn’t apply, but you could also say that every point in the circuit is a junction point and whatever charge enters the point leaves the point, so that shows that the current must be the same everywhere.

Question 7

Resistance in series

Answer 7

Rt = R1 + R2 + R3 (or however many resistors there are)

→ This is very logical - since every charge must pass through all of the loads as it travels from the positive terminal of the battery through the circuit to the negative terminal, the total opposition is the sum of the oppositions provided by each of the loads.

• The total resistance of a set of resistors connected in series is the sum of the individual resistances, and so will always be more than the largest individual resistance.

Question 8

Note about when loads are in series:

Answer 8

the potential differences across them are proportional to the resistances, so the one with greater resistance has a greater proportion of the total potential difference across it.

Question 9

How does electrical power combine in a series circuit?

Answer 9

Since power is the rate at which energy is converted, and each load is converting energy at the same time, the total power is simply the sum of the individual powers

Pt = P1 + P2 + P3

Question 10

Parallel circuits:

Answer 10

Components are connected so that the current splits along different branches

Question 11

Voltage in Parallel:

Answer 11

The potential difference across each resistor is the same and is equal to the potential difference produced by the battery.

Vt = V1 = V2 = V3

How does this relate to Kirchoff’s voltage Law?

Since each charge passes through only one of the loads, as it goes from the positive terminal of the battery to the negative terminal, all of the electrical energy it was given by the battery is converted by that one load, so the potential difference across each load is the same as that produced by the battery. th battery and each individual lead make up a complete path.

Question 11

Current in Parallel:

Answer 11

each electron passes through only one of the loads. the amount of current through each depends on what the resistance of each load is. So the total circuit current is the sum of the currents through the individual resistors.

It = I1+ I2 + I3

This agrees with Kirchoffs current law

Question 12

Resistance in Parallel:

Answer 12

The total resistance of a set of resistors connected in parallel is alwsy less than the smallest individual resistance

this sis logical because each time we add a resistor, we provide another path for current to flow from the battery back to the battery, meaning less overall resistance to the flow

1/Rt = 1/R1 + 1/R2 + 1/R3

or

Rt = (1/R1 + 1/R2 + 1/R3) ^-1

Question 13

One of the resistors in the above problem is replaced by a 10.0-2 resistor.

a) Does Rt change? If so, how?

b) Does It change? If so, how?

c) Does the current through the other two resistors change? If so, how?

Answer 13

A) decreases

B) Increases

C) No because the current through each depends on their resistance and the voltage across them, neither of which changed.

Question 14

How does electrical power combine in a parallel circuit?

Answer 14

Since power is the rate at which energy is converted, and each load is converting energy at the same time, the total power is simply the sum of the individual powers

Pt = P1 + P2 + P3

Question 15

Combination Circuits

Answer 15

When possible we use the rules for series and parallel combinations to determine the total resistance of the circuit, then find the total current and go from there.

Question 16

Emf and terminal voltage

Answer 16

• When current is drawn from a battery, the voltage across its terminals drops below its rated emf.
• This is because the chemicals and electrodes in the battery cannot supply charge fast enough. There is some resistance to the flow of charge inside the battery.

Question 17

The internal resistance of a battery

Answer 17

is the resistance of the battery itself. (The symbol used for this is normally r rather than R) → This resistance is generally very low, but it can matter. It cannot be separated from the battery.

Question 18

The emf of a battery

Answer 18

is the potential difference maintained across the battery when no current is flowing.

Question 19

The terminal voltage of a battery (Vab or Vterm)

Answer 19

is the potential difference across the terminals of the battery at any time.
→ When no current is flowing, the terminal voltage will be the same as the emf.
→ When current is flowing, there is a drop in voltage across the terminals of an amount Ir.
* So the terminal voltage of the battery is given by Vterm = ℰ - Ir

Question 20

Emfs in series and parallel

Answer 20

When emfs are in series, the voltages add algebraically. → e.g., most small devices such as CD players or flashlights have several 1.5 V batteries in series. If there are two 1.5-V batteries in a series, it is equivalent to a 3.0-V battery.

• If the batteries are in opposite directions, the voltages will subtract. (If they are not equal in voltage, the net voltage will simply be the difference between the two.)
• When emfs are in parallel, the voltages do not add, but the current capacity is increased. (because there is less current through each battery and so there is less loss to internal resistance).

Question 21

Note about when a battery is being charged

Answer 21

current is being forced through it in the opposite direction than it would normally flow. In this case, the terminal voltage is actually higher than the emf, and the appropriate formula is actually V = ℰ+lr.