Solution:

a) Charge q as a function of time:

The charge q on the capacitor can be expressed as:

q = C * V

where C is the capacitance of the capacitor and V is the voltage across the capacitor.

Since the capacitor is being charged by a DC source, the voltage across the capacitor will increase linearly with time until it reaches the voltage of the source.

The voltage across the capacitor can be expressed as:

Vc = E * (1 - e^(-t / (R * C)))

where E is the voltage of the DC source, t is the time, R is the resistance of the resistor, and C is the capacitance of the capacitor.

Using the above equation, the charge on the capacitor can be expressed as:

q = C * E * (1 - e^(-t / (R * C)))

b) Charging current:

The charging current can be expressed as:

I = V / R

where V is the voltage across the resistor.

Since the voltage across the capacitor is increasing linearly, the voltage across the resistor will decrease linearly until it reaches zero.

The voltage across the resistor can be expressed as:

Vr = E * e^(-t / (R * C))

Using the above equation, the charging current can be expressed as:

I = E / R * e^(-t / (R * C))

c) Charging voltage:

The charging voltage is the voltage across the capacitor, which is given by:

Vc = E * (1 - e^(-t / (R * C)))

d) Voltage across resistor:

The voltage across the resistor is given by:

Vr = E * e^(-t / (R * C))

e) Voltage across capacitor:

The voltage across the capacitor is given by:

Vc = E * (1 - e^(-t / (R * C)))

f) Plot all the waveforms:

The waveforms can be plotted as follows:

Q vs t: A plot of charge on the capacitor vs time. The graph will start from 0 and increase linearly until it reaches the maximum charge on the capacitor.

I vs t: A plot of current vs time. The graph will start from its maximum value and decrease exponentially until it reaches 0.

Vc vs t: A plot of voltage across the capacitor vs time. The graph will start from 0 and increase linearly until it reaches the voltage of the DC source.

Vr vs t: A plot of voltage across the resistor vs time. The graph will start from the voltage of the DC source and decrease exponentially until it reaches 0.

Note: It is recommended to use appropriate software or tools to plot these waveforms accurately.

Conclusion:

In this question, we have derived expressions for various parameters like charge, current, voltage across the capacitor, and voltage across the resistor for a capacitor in series with a resistor connected to a DC source. We have also plotted the waveforms for each parameter. These waveforms help us to understand the behavior of the circuit during the charging process.