Question

In this chapter we are looking at the differential equations that govern RC circuits. For a capacitor that is discharging, we know that going around the circuit the voltage should sum to zero:

Vc + Vr = 0.

Written in terms of charge and current, this is:

q/C = - i R.

Rewritten, q/C = -dq/dt R.

What this is telling us is that the current is being set by the charge on the capacitor. We can use a numerical approach to model this system. We know that dq/dt is a change in charge divided by a change in time:

Δq/Δt or (qf-qi)/Δt, where qi is the initial charge and qf is the final charge.

Treating q as qi, this allows us to state:

qi/C = -(qf-qi)/Δt.

Solving for qf, we get:

qf = qi – (qi/RC) Δt.

qf = qi (1 – Δt/RC).

In other words, at each small step in time, we subtract off a value proportional to the current charge.

We can model this behavior in a spreadsheet. We’ll use some tricks to help us out. In the top of the spreadsheet we’ll define four quantities: Δt, R, C and qi. This gives us details of the circuit and also the change in time. I put in some sample values, feel free to change them in order to experiment with how circuits work.

We then set up three columns. The first is a time column, the second is qi and the third is qf. Let’s look at first row. Time initial is zero, charge initial is set by the variables at the top of the chart and qf is done by formula, it is qi multiplied by (1–Δt/RC). Note that these values use absolute relationships, eg. $B$2, so if the formula gets copied, then it always refers back to the values at the top of the chart.

The next line down is really the key to the whole affair. The time column is the time from the line above, plus the delta t. The qi is the qf from the previous line, and the qf is calculated from the qi on this line. Note that since so many of the entries are in reference to the previous line, we can simply copy this expression to the line below to now have a three-step process. In fact, we can do this repeatedly (I did it 500 times) to watch how q changes from step-to-step in time. I graphed the q(t) behavior, and you can see it is an exponential, just as predicted by theory.

For this week’s homework, add a battery to this circuit and make a new spreadsheet that corresponds to charging up an empty capacitor. Your spreadsheet should include the appropriate graph. Note that for things like RC circuits, it is usually simpler to just solve the differential equation. For more complex problems, scientists and engineers will often use numerical methods rather than directly tackle hard math problems.

Answer 1

For the first part of question, i.e. discharging capacitor through resistance,

procedure of getting charge on capacitor plates a s a function of time is clearly explained.

Hence I do the homework problem, i.e., charging capacitor through a resistance using a battery

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Let us assume a charging circuit as shown in figure.

when key is closed, after some time t, let i be the charging current, q be the charge on capacitor.

Then we have , E = i R + (q/C)

if we substitute i = dq/dt, then we have , E = (dq/dt) R + (q/C)

let qbe the charge at time t, then difference equation for above differential equation is written as

Above equation can be rearranged as,

Incremental charge added at each time interval depends on previous charge present on the capacitor plates in addition to EMF of battery, resistance R and capacitance C of capacitor.

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Excel chart is given below

	A	B	C	D	E	F	G
	time (ms)	delQ (C )	Q ( C)	delt(ms)	E ( V )	R ( ohm )	C (uf)
	0	0	0	5	5	470	100
	5	5.31915E-05	5.32E-05
	10	4.75328E-05	0.000101
	15	4.24761E-05	0.000143
	20	3.79574E-05	0.000181
	25	3.39194E-05	0.000215
	30	3.03109E-05	0.000245
	35	2.70864E-05	0.000272
	40	2.42048E-05	0.000297
	45	2.16299E-05	0.000318
	50	1.93288E-05	0.000338
	55	1.72726E-05	0.000355
	60	1.5435E-05	0.00037
	65	1.3793E-05	0.000384
	70	1.23257E-05	0.000396
	75	1.10144E-05	0.000407
	80	9.84269E-06	0.000417
	85	8.79559E-06	0.000426
	90	7.85989E-06	0.000434
	95	7.02373E-06	0.000441
	100	6.27653E-06	0.000447
	105	5.60881E-06	0.000453
	110	5.01213E-06	0.000458
	115	4.47892E-06	0.000462
	120	4.00244E-06	0.000466
	125	3.57665E-06	0.00047
	130	3.19616E-06	0.000473
	135	2.85614E-06	0.000476
	140	2.55229E-06	0.000479
	145	2.28077E-06	0.000481
	150	2.03814E-06	0.000483
	155	1.82132E-06	0.000485
	160	1.62756E-06	0.000486
	165	1.45441E-06	0.000488
	170	1.29969E-06	0.000489
	175	1.16142E-06	0.00049
	180	1.03787E-06	0.000491
	185	9.27457E-07	0.000492
	190	8.28791E-07	0.000493
	195	7.40622E-07	0.000494
	200	6.61832E-07	0.000494
	205	5.91425E-07	0.000495
	210	5.28507E-07	0.000496
	215	4.72283E-07	0.000496
	220	4.2204E-07	0.000496
	225	3.77142E-07	0.000497
	230	3.37021E-07	0.000497
	235	3.01167E-07	0.000497
	240	2.69128E-07	0.000498
	245	2.40498E-07	0.000498
	250	2.14913E-07	0.000498

for delQ, formula used at B3 :- =($E$2 * $G$2 * 1.0e-06 - C2) * ( $D$2 * 1000 / ( $F$2 * $G$2 ) )

for Q , formula used at C3 :- =(C2+B3)

Plot of charge as a function of time is given below