A 100 capacitor is joined to a 24 V battery via a 1.0 M resistor....

The given circuit is an RC charging circuit.

The time constant of the circuit is given by $\tau = RC$.

Substituting the given values:

$\tau = (1.0 \times 10^6 \Omega) \times (100 \times 10^{-6} \text{ F}) = 100 \text{ s}$

The initial current in the circuit at $t = 0$ is:

$I_0 = \dfrac{V}{R} = \dfrac{24}{1.0 \times 10^6} = 24 \mu\text{A}$

The current in an RC circuit during charging varies with time according to the equation:

$I(t) = I_0 e^{-t/\tau}$

$I(t) = 24 e^{-t/100} \mu\text{A}$

This equation represents an exponentially decaying curve. At $t = 0$, the current is $24 \mu\text{A}$, and as time progresses, it exponentially approaches zero.

For a duration of $10$ minutes ($600 \text{ s}$, which is $6\tau$), the current will have decayed to a value very close to zero. Thus, the graph is an exponentially decaying curve starting from $24 \mu\text{A}$ and approaching zero.