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When recharging a lead-acid battery, how are the terminals of the battery connected to the voltage source. I mean, do we connect the positive terminal of the battery to the negative terminal of the source and vice versa? Or does the positive go with the positive and the negative with the negative? Also, are the electrodes reversed while the battery is being recharged?

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The polarities are not reversed, basically current always flows from the positive to negative terminal (opposite the flow of electrons by definition). When a battery is discharged, the voltage difference across its terminals will drop. When you then connect the charger with the full voltage across the terminals of the batteries, the chemical process that lead to the discharge of the battery is reversed and the current flows "backwards" if you will, with the help of the charger. So you could think of the battery being discharged a little, and having only 10 volts instead of 12 volts across its terminals, connecting the 12 volts charger to those terminals (negative to negative and positive to positive) means that there's a voltage difference of 2V from the positive terminal of the charger to the positive terminal of the battery which causes the current to flow and the battery to charge.
You should look at redox processes if you want to understand more of the chemistry behind it! If you just want to charge your battery, remember to connect the positive terminal of the charger to the positive terminal of the battery!

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The voltage on the terminals is different by 12 V (± a few depending on the charge state), you can think of this as the positive terminal being at +12 V and the negative terminal at being 0 V. The polarity of the voltage does NOT change.

On the current, things are a bit different. During discharge, the current flows from the positive to the negative.

During charge, the current flows out of the positive terminal of the charger into the positive terminal of the battery (and out of the negative terminal of the battery into the negative terminal of the charger).

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