Internal resistance and Kirchhoff's laws

Key ideas

• A real battery is a perfect EMF source in series with a small internal resistance $r$ inside the battery itself.
• The EMF $\varepsilon$ is the total energy the battery gives each coulomb. Some of it is spent pushing charge through the battery's own $r$, so the terminal voltage — what the external circuit actually receives — is less when current flows:

$\varepsilon = I(R + r) \qquad V_{terminal} = \varepsilon - Ir$

• Each variable, with sub-bullets:
  • $\varepsilon$ — the EMF of the battery (V)
  • $r$ — the internal resistance (Ω)
  • $R$ — the resistance of the external circuit (Ω)
  • $I$ — the current (A)
• The larger the current drawn, the more voltage is "lost" inside the battery — which is why headlights dim when the starter motor runs.

Kirchhoff's laws

• Junction (current) law: the total current into a junction equals the total current out — charge is conserved, it can't pile up.
• Loop (voltage) law: around any closed loop, the sum of the EMFs equals the sum of the $IR$ voltage drops — energy per coulomb is conserved.
• Method for a circuit question:
  • Label every current with a name and an assumed direction.
  • Apply the junction law at a junction.
  • Apply the loop law to enough loops to match your unknowns.
  • A negative answer just means the true current flows opposite to your arrow.

Tips

• Measure EMF with no current flowing. With the circuit open, $I = 0$, so $V_{terminal} = \varepsilon$ — a voltmeter across an open battery reads the EMF. Under load it always reads less.
• In loop-law working, go around the loop one way and keep signs consistent: EMFs positive when you pass − to +, drops positive in the direction of current. Mixing conventions mid-loop is the classic error.