A battery pushes charge round a loop. That much everyone agrees on. The argument starts one step later, over what happens to the charge on the way.

One current round the loop#

Ohm's law, and the two rules that do the work#

For one resistor, Ohm's law: $V = IR$. For the whole circuit, Kirchhoff's two rules, both conservation laws:

$$\sum_{\text{loop}} \Delta V = 0 \qquad\qquad \sum_{\text{junction}} I_{\text{in}} = \sum_{\text{junction}} I_{\text{out}}$$

Round any loop the voltages add to zero: energy. At any junction what flows in flows out: charge.

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Adding a branch#

The battery does not supply a fixed current#

A battery is not a tank of current. It maintains an EMF, a fixed voltage, and the current is whatever that voltage and the circuit produce together. A real battery has resistance of its own, $r$, so the voltage at its terminals is

$$V_{\text{terminal}} = \mathcal{E} - I r$$

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Charging a capacitor: current that dies, charge that grows#

Charge on one plate pushes charge off the other, and nothing ever crosses the gap. As the plates charge, their voltage opposes the battery, so the current falls:

$$q(t) = C\mathcal{E}\left(1 - e^{-t/RC}\right) \qquad I(t) = \frac{\mathcal{E}}{R}\,e^{-t/RC}$$

$RC$ is the time constant. At $t = RC$ the capacitor holds $1 - e^{-1} = 63.2\%$ of its final charge.

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What a capacitor stores#

While the capacitor charges, the battery hands over $CV^2$ of energy. Half of it leaves as heat in the resistance, however large or small that is. The other half is the $\tfrac{1}{2}CV^2$ the capacitor is left holding, in the field between its plates.

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What comes next#

You counted one current round a loop, added a branch, loaded a real battery and timed a capacitor, all with Ohm's law and two conservation rules. Next, a moving charge in a magnetic field: the force that turns a current into motion.