Current Division


Analogous to voltage dividers is the current divider.

When there are parallel branches in a circuit they all share the same voltage by definition. The resulting current depends on the ratio of the resistances.

The common saying that current takes the path of least resistance is a misnomer.
Current actually takes all paths simultaneously, with inverse proportion to the branch resistance. It’s definitely not as easy to say!
A saying that would be true would be to say that current takes the path of zero resistance, if zero resistance existed.

Analysis

Parallel branches share a common voltage. In a resistive circuit therefore each branch will carry current:
$$I_n = \frac{V}{R_n}$$
The total current will then be:
$$I = I_1 + … + I_n = \frac{V}{R_1} + … + \frac{V}{R_n}$$
$$I = \frac{V}{R_{parallel}} + \frac{V}{R_k}$$

Each branch can then be written as:
$$I_k = I \frac{1}{1 + \frac{R_k}{R_{parallel}}}$$

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