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If the emf of a battery is E Volt, the potential difference across a battery is given by
V = E -I r where I is the current in the circuit and r is the inetrnal resistance.

Hence E and V will be equal only when I = 0.

The maximum potential difference across the battery will be equal to E only if I = 0.

In gnereral potential difference can be equal or less than the emf.

E.m.f can never exceed the potential difference.
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A battery charger is a device used to put energy into a secondary cell or (rechargeable) batteryby forcing an electric current through it.

Hence to charge a battery another source of emf is needed.

The combined emf is now will be (E - E1) where E is the emf of the battey in quesiton and E 1 is the emf of the external source used to charge the battery.

Note that E-E1 will be negative in sign.

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A battery is charged only when its emf is less than its maximum emf.

Suppose that the maximum emf of a cell is 1.5V. The battery should be charged only when its emf is less than 1.5 V say 0.5 V.

To charge the cell we use a different source of emf E1 say 3V.

The positive of the second source is connected to the negative of the cell so that the
Combined emf is now 0.5 - 3 = -2.5V.

The negative sign indicates that the emf is opposite to the emf of the cell which is 0.5V.

Since the cell is getting charged, the difference in emf is gradually reduced to zero when the cell is fully charged.

In modern charging units there are provisions so that the cell is never allowed to be over charged, even if the charging unit is in on for about 12 hours.

When the cell is fully charged, (that is when the emf of the cell is now 1.5V), the potential difference between either the second source or cell will be zero.

Taking into consideration the sign of the emf and the direction of current through the cell and the sign of the potential difference, the potential difference will be always less than the emf of the cell (which gradually increases while charging).

Note that the potential difference is negative if the emf of the cell is taken as positive.

Also note that the cell is charged only when its emf is less than its maximum e.m.f

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Q: Can the potential difference across a battery be greater than its EMF?
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