Topics: DC current, Combine resistors, and
Kirchhoff’s Circuit Rules
Related Reading: Lea textbook, Chapter 26. 3 till the end of Chapter 26
Related HW : HW 4c, 4d .
When resistors are in series, the following are true:
The total effective resistance is the sum of all the resistance. R total
= R
1
+R
2
+R
3
+… The total R is more than any of the individual resistance.
The current through each resistor is the same as the current though the battery. I= I
1
=I
2
=I
3
…
I=

V/( R
1
+R
2
+R
3
+…)= 
V/R total
Here,

V is the voltage drop across all these resistors.

V is not the voltage drop crossing one resistor.
Actually, the total

V= IR
1
+IR
2
+IR
3…
  
V=

V
1
+

V
2
+

V
3…
The total voltage drop is the sum of the voltage drop across each individual resistor.
For any one of the resistor (the ith resistor, R i
), its voltage drop

V i
= R i
I= R i

V /(R
1
+R
2
+R
3
+…).
This is called voltage divider. Each resistor shares a portion of the voltage drop
∆𝑉 𝑖
= ∆𝑉
𝑅 𝑖
R1 +R2+R3+⋯
= ∆𝑉
𝑅 𝑖
R total
When you consider the power consumed by each resistor when they are in series, P=I
2
R may be a good equation to use, since I is the same for all resistors and easy to calculate. If you want to use P=

V
2
/R to calculate the power consumed by one resistor, make sure that you are using

V across that part of resistor, instead of using the total

V from the battery or the

V for the entire series of resistor.
When you consider the power provided by the battery, P=I

V, may be a good equation to use. All you need is to find the current through the battery. If you use P=

V 2 /R or P=I 2 R to calculate the power provided by the battery, you need to make sure that you are using the total effective R, not R for one or part of the resistors.
When resistors are in parallel , the following are true:
We need to add those resistors “ reversely” to calculate the total effective resistance.
The total resistance is less than any individual resistance. Because when resistors are in parallel, there is always extra channel for current to flow, besides any individual path.
The voltage drop through each path (each resistor) is the same

V.
For any one of the resistor (the ith resistor, R i
), its current
,
I i

V/ R i
The current through each resistor is not the same. Those large resistors will get a smaller current.
The total current going through the battery and the main path (before the current splits) is equal to the sum of the current through all the paths.

𝐼 𝑡𝑜𝑡𝑎𝑙 total
=
= I
1
+

𝑅
∆𝑉 𝑡𝑜𝑡𝑎𝑙
2
+I
3
+
…
1
= ∆𝑉 (
𝑅
1
+
1
𝑅
2
+
𝑅
1
13
+ ⋯ )
Here,

V is the voltage drop through each path, as well as the total

V.
The total current is the sum of the current through each individual resistor. You can consider it as “current divider.” If any branch is shorted and has zero resistance, its current will become infinite and current won’t

flow into other branches. The battery will be burn right away due to the huge amount of power associated with the instance infinite current.
When you consider the power consumed by each resistor in parallel, P=

V 2 /R may be a good equation to use, since

V is the same for branches. If you want to use P=
 2
R to calculate the power consumed by one resistor, make sure that you are using the actual current across that resistor, instead of using the total current (the total current going through the battery).
When you consider the power provided by the battery, P=I total

V is still a good equation to use. All you need is to find the Total current through the battery. If you use P=

V 2 /R or P=I 2 R to calculate the power provided by the battery, you need to make sure that you are using the total effective R, not R for one or part of the resistors.
Notice that the total R of the parallel resistors is less than any of the individual R before they were put in parallel, the total power provided by the battery will become more when you add resistors in parallel.
Kirchhoff’s Circuit Rules
In analyzing circuits, there are two fundamental (Kirchhoff’s) rules: (1) The junction rule states that at any point where there is a junction between various current carrying branches, the sum of the currents into the node must equal the sum of the currents out of the node (otherwise charge would build up at the junction);
(2) The loop rule states that the sum of the voltage drops Δ across all circuit elements that form a closed loop is zero (this is the same as saying the electrostatic field is conservative).
If you travel through a battery from the negative to the positive terminal, the voltage change is Δ V= +ε, because you are moving against the internal electric field of the battery. Voltage at the positive terminal is higher than the voltage at the negative terminal.
If you travel through a battery from the negative to the positive terminal, the voltage change is Δ V=-
ε.
If you travel through a resistor in the direction of the assumed flow of current, the voltage change is –IR , because you are moving parallel to the electric field in the resistor;
If you travel through a resistor opposite to the direction of the assumed current flow, the voltage change is
Δ
V=+IR .
If later the current I is solved to be negative, it means the actual direction of the current flow is opposite to what was assumed.
Steps for Solving Multi-loop DC Circuits
1) Draw a circuit diagram, and label all the quantities;
2) Assign a direction to the current in each branch of the circuit--if the actual direction is opposite to what you have assumed, your result at the end will be a negative number;
3) Apply the junction rule to the junctions;
4) Apply the loop rule to the loops until the number of independent equations obtained is the same as the number of unknowns.
Safety:
The human body resistance varies between 300 to 1000 Ohms. (For example, when hands are wet, your resistance becomes much smaller). When Voltage difference cross the body is less than 36 Volt, it is always safe. Otherwise, the Voltage difference can generate a large enough current to hurt the body. It is still safe to stand on top of a good isolator before touching higher voltage. Your conducting body can soon have the same potential (voltage), without voltage difference and without current going though. The high voltage difference will be between the two ends of the good isolator.