An introduction to electrical circuits, and their use in physiology
Resistors in parallel
When current is obliged
to pass through all the resistors in a circuit, one after the other,
the resistors are said to be in series, as in the previous examples.
But it is also possible to design a circuit in which current has
a "choice" of directions to take, whereby some of the current need
not pass through a given resistor on its way back to the battery.
In this case, as in the diagram shown, the two resistors are said
to be in parallel.
In the diagram, the total current in the circuit, I, divides: some of it (I1) travels through the resistor R1, whereas the rest (I2) travels through the resistor R2, such that
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I = I1 + I2 |
(6) |
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Now, the potential at any place in the wire on the left of the circuit takes the same, positive value, and the potential at any place on the wire on the right takes the same, negative value (the fact that the wire splits is irrelevant: there's nothing about the wire itself which would cause the potential to change). This means that the voltage V, recorded by the voltmeter across the resistor R1, as shown, must be the same as the voltage across the battery (i.e. 3V if we continue to use the same battery as before). We would record the same voltage across the resistor R2. In fact, the same applies no matter how many resistors are placed in parallel: the voltage across each of the resistors arranged in parallel must be the same. This is in contrast to the case in the series circuit considered earlier, where the total voltage was divided among the various resistors. We can use this fact, in conjunction with Ohm's law (V = IR), to work out how much current passes through each resistor. Considering first the upper and then the lower branches of the circuit: |
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V = I1R1 V = I2R2 |
(7) (8) |
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By combining these equations: |
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and therefore |
I1R1 = I2R2 I1/I2 = R2/R1 |
(9) (10) |
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The ratio of the two currents is therefore the reciprocal of the ratio of the two resistances that they pass through. If we use the relationship I2 = (I – I1) from Equation (6), and do some rearranging, we can easily derive the following: |
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I1 = I.R2 / (R1 + R2) |
(11) |
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Translating this into English, the lower the resistance R1 is relative to R2, the greater the fraction of the total current which takes the upper branch of the circuit, this route being relatively easier. If the resistance R2 is infinite, no current will pass this way and instead all current will pass through R1... the parallel circuit, in effect, has become a simple series circuit. Conversely, if the resistance R2 is zero, all the current will take this route and none will pass through the upper branch – this is known as a short-circuit.
If there are other physics-related areas that you would like to see introduced in a similar way on-line, please contact Dr Matt Mason