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EMF and Terminal Voltage

Terminal voltage V equals EMF ε minus voltage drop across internal resistance r during discharge. During charging, V = ε + Ir — external source must exceed EMF.
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Derivation

EMF defined

EMF E\mathcal{E} is the work done per unit charge by the non-electrostatic force (chemical, electromagnetic) inside the source in driving charge from the negative to positive terminal:

E=Wnon−electrostaticq\mathcal{E} = \frac{W_{non-electrostatic}}{q}

During discharge (cell drives current)

Applying KVL around the loop (cell + external resistance RR):

E=IR+Ir=I(R+r)\mathcal{E} = IR + Ir = I(R + r)

Terminal voltage (potential difference across the external circuit):

V=E−Ir\boxed{V = \mathcal{E} - Ir}

V<EV < \mathcal{E} during discharge — internal resistance causes a voltage drop.

During charging (external source drives current into cell)

Current is forced in the reverse direction through the cell. KVL gives:

V=E+Ir\boxed{V = \mathcal{E} + Ir}

V>EV > \mathcal{E} during charging — external source must exceed EMF to push charge in.

Open circuit

I=0⇒V=EI = 0 \Rightarrow V = \mathcal{E}. Terminal voltage equals EMF only when no current flows. A voltmeter (very high resistance) measures close to E\mathcal{E}.

Remember
A battery with low internal resistance $r$ maintains terminal voltage close to $\mathcal{E}$ even under heavy load. A "weak" battery has high $r$ — its terminal voltage drops significantly under load.