Resistors in series
When resistors are in series they are arranged in a chain, so the current has only one path to take and is therefore the same through each resistor.
The sum of the potential differences across each resistor equals the total potential difference across the whole chain. For two resistors in series we get:
DV = DV1 + DV2
I Req = I R1 + I R2
Because the current is the same, we get:
Req = R1 + R2
This is true in general, and can be extended to any number of resistors. The equivalent resistance of resistors in series is:
Req = R1 + R2 + R3 + ...
The single equivalent resistor has the same current through it as each resistor in the series circuit, and the potential difference across it equals the total potential difference across the whole chain of resistors. A battery can't tell the difference between the series chain of resistors or the equivalent resistor.
Resistors in parallel
When resistors are arranged in parallel, the current has multiple paths to take. In parallel the resistors are all connected together at one end, and are also all connected together at the other end. The potential difference across each resistor is the same, and the currents add to equal the total current entering (and leaving) the parallel combination.
For two resistors in parallel:
I = I1 + I2.
The potential differences are all the same, so:
This is true in general, and can be extended to any number of resistors. The equivalent resistance of resistors in parallel is:
| 1
|  |
| Req
|
|
= |
| 1
| |
| R1
|
|
+ |
| 1
| |
| R2
|
|
+ |
| 1
| |
| R3
|
|
+ ... |
Series example
Three resistors, with values of 8 W, 8 W,
and 4 W are connected in series to a 10-volt battery.
(a) What is the total current provided by the battery?
(b) What is the potential difference across each resistor?
First find the equivalent resistance, which is 20 W, the sum of the individual resistances.
The current from the battery is:
| I |
= |
| DV
| |
| R
|
|
= |
| 10
| |
| 20
|
|
= |
0.5 A |
This is the current passing through each resistor. The potential difference across each resistor can by found using Ohm's Law:
Each 8 W resistor has a potential difference DV = I R = 4 V
The 4 W resistor has a potential difference DV = I R = 2 V
The sum of the potential differences across each resistor equals the battery voltage, as it should.
Parallel example
Three resistors, with values of 8 W, 8 W,
and 4 W are connected in parallel with one another and with a 10-volt battery.
(a) What is the total current provided by the battery?
(b) What is the power dissipated in each resistor?
First find the equivalent resistance, which is:
| 1
| |
| Req
|
|
= |
| 1
| |
| R1
|
|
+ |
| 1
| |
| R2
|
|
+ |
| 1
| |
| R3
|
|
= |
| 1
| |
| 8
|
|
+ |
| 1
| |
| 8
|
|
+ |
| 1
| |
| 4
|
|
= |
| 4
| |
| 8
|
|
= |
| 1
| |
| 2
|
|
Flip this upside down to get Req = 2 W
| I |
= |
| DV
| |
| Req
|
|
= |
| 10
| |
| 2
|
|
= |
5 A |
The current through each resistor can be found using Ohm's Law.
| For each 8 W resistor, I |
= |
| DV
| |
| R
|
|
= |
| 10
| |
| 8
|
|
= |
1.25 A |
| For the 4 W resistor, I |
= |
| DV
| |
| R
|
|
= |
| 10
| |
| 4
|
|
= |
2.5 A |
The sum of the currents equals the total current from the battery, as it should.
The power dissipated in each resistor can be found a number of different ways. Here's one method:
| For each 8 W resistor, I |
= |
| DV2
| |
| R
|
|
= |
| 10*10
| |
| 8
|
|
= |
12.5 W |
| For the 4 W resistor, I |
= |
| DV2
| |
| R
|
|
= |
| 10*10
| |
| 4
|
|
= |
25 W |
That's a total of 50 W. Check this against the power provided to the circuit by the battery:
P = DV I = 10 * 5 = 50 W.
They agree, as they should.