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Mains Power

why are transformers used instead of potential dividers used to reduce the mains power down to a lower level?

There are two reasons:
1) because potential dividers have low efficiency (considerable power dissipation).
2) because output voltage changes with the load current.

Let's suppose that potential divider consists of two resistors R1 and R2 so that R2 is parallel to load.
If we use small value for R2, then significant power is dissipated through R2 and we have problem (1).
If we use large value for R2, then most of the current drawn by the load that passes through resistor R1, doesn't pass through R2, causing the voltage drop across R1 to change with the load current. This leads to problem (2).

Only advantage of potential dividers is that may be used to produce precise voltages in cases when very small current is drawn from the reference node. Because of this, they are used as voltage regulators in low-power electronics. For example, classic design of volume regulation in audio devices consists of potentiometer used as variable potential divider.

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