2018 A/L Physics Past Paper · Question 21 (English Medium)

Answer: (3) 5 μC

What the question asks

We want: the total charge on the three capacitors when they are fully charged.

$$Q = \Sigma\,C\,\Delta V$$

Given information

Each capacitor is $1\ \mu\text{F}$. The cell gives 10 V.

There are three branches across the cell.

The left branch and the middle branch each have five equal $R$.

The right branch has two equal $R$.

Relevant theory

The capacitance of a capacitor can be considered as a measure of its ability to store electric charges.

When a certain fraction only of an input voltage is required to be used for another electrical activity, the circuit set up called the "potential divider" is used.

Reasoning and calculation

When the capacitors are full, no current flows through them.

So current flows only down the three branches.

Each branch is a potential divider across the full 10 V.

The left branch and the middle branch are the same.

So points at the same level in them are at the same potential.

Take the bottom wire as 0 V.

In a branch of five $R$, each $R$ takes 2 V.

So going down, the points are at 8 V, 6 V, 4 V and 2 V.

The middle of the right branch is at 5 V.

The middle capacitor joins the 6 V points of the left and middle branches.

Both its ends are at 6 V. So it has no charge.

The left capacitor is across one $R$ of the left branch. It has 2 V.

The right capacitor joins the 8 V point to the 5 V point. It has 3 V.

Required equations

$I = \frac{10}{5R} \Rightarrow IR = 2\ \text{V}$

$V = 10\times\frac{R}{2R} = 5\ \text{V}$

$Q = C\,\Delta V$

$2$

$Q_1 = 1\,\mu F \times 2\ V$

$0$

$Q_2 = 1\,\mu F\times(6 - 6)$

$3$

$Q_3 = 1\,\mu F\times(8 - 5)$

$5$

$Q = 2 + 0 + 3$

Reading the options

The total charge is 5 μC.

That is option (3).

Checks

$C * (2 + 0 + (8 - 5))$

Check: the left branch and the middle branch are the same.

So for the middle capacitor

$$\Delta V = 0$$

It holds no charge.

Conclusion

$Q = 5\ \mu\text{C}$

$$Q = 2\,\mu\text{C} + 0 + 3\,\mu\text{C} = 5\,\mu\text{C}$$

The answer is option (3).