2022 A/L Physics Past Paper · Question 31 (English Medium)

Answer: (3) Only (A) and (C) are true.

What the question asks

What we know:

One metal wire has a thick section, a thin section and another thick section, joined one after the other.

A steady current flows along the whole wire.

The question gives no lengths and no numbers.

We want: which of the statements (A), (B) and (C) are true.

What each letter means:

$I$ ($\text{A}$): the current in the wire.

$v$ ($\text{m s}^{-1}$): the drift velocity of the electrons.

$A$ ($\text{m}^2$): the area of cross-section.

$n$ ($\text{m}^{-3}$): the number of free electrons in one cubic metre of the metal.

$e$ ($\text{C}$): the charge of an electron.

$R$ ($\Omega$): the resistance of a section, $l$ ($\text{m}$) its length and $\rho$ ($\Omega\ \text{m}$) the resistivity of the metal.

$P$ ($\text{W}$): the power turned into heat in a section.

Relevant theory

What is happening:

The free electrons in the metal drift along the wire all the time while the current flows.

Charge cannot pile up anywhere in the wire. So every second, the same number of electrons crosses every cross-section, thick or thin.

In the thin part there is less room. To get the same number through each second, the electrons there must move faster.

The electrons bump into the metal atoms as they drift, and that makes heat. A metre of the thin part has more resistance than a metre of the thick part, so it gets hotter.

An electric current is measured by the rate at which charges flow across a cross-section of the conductor through which it is flowing.

The velocity of the electrons of an electric current flowing through a conductor is called the "drift velocity" of the electrons.

The constant p represents the contribution from the material of the conductor to its resistance and is known as the "resistivity" of the material.

Reading the options

The options are combinations of (A), (B) and (C). We test each statement on its own.

The current is the charge crossing a cross-section each second. For electrons drifting with speed $v$:

$$I = vAne$$

In words: the current is the drift velocity times the area times the number of electrons per cubic metre times the charge of one electron.

The resistance of a section and the power turned into heat in it are:

$$R = \dfrac{\rho l}{A}$$

$$P = I^2R$$

In words: a thinner or longer section has more resistance, and more resistance with the same current gives more heat.

For example, a crowd leaves a stadium through a wide gate, then a narrow gate, then another wide gate. The same number of people pass each gate every minute, but at the narrow gate they have to walk faster.

The electrons are like that crowd, and the thin section is the narrow gate.

Reasoning and calculation

The current in the wire is same everywhere.

The current is the same in every section. Charge cannot pile up, so the charge crossing each cross-section per second is the same.

A

The drift velocity of electrons is higher in the thin section than that in thick section.

With the same $I$, a smaller $A$ needs a bigger $v$. From $v = \dfrac{I}{nAe}$, the electrons drift faster in the thin section.

C

The power dissipation of the wire is higher in the thin section than that in thick section.

A metre of the thin section heats more than a metre of the thick section, because its resistance per metre is bigger. But the statement compares whole sections, and their lengths are not given. A long thick section can have as much resistance as a short thin one. So (B) is not always true.

B

Statements (A) and (C) are true.

Statement (B) holds only per unit length, and the lengths are not given.

So only (A) and (C) are true.

Why the other options fail

Statement (C) is also true: the electrons drift faster in the thin section.

Statements (A) and (C) are true, and (B) depends on the lengths, which are not given.

Statement (A) is true: the current is the same in every section.

Statement (B) holds only for equal lengths, and the lengths of the sections are not given.

Conclusion

Only statements (A) and (C) are true. (B) holds only per unit length.

The answer is (3).

In a wire in series the current is the same in every part. Then $I = vAne$ tells you at once that a thinner part has a faster drift.

For heat, look at the resistance of the whole section, $R = \dfrac{\rho l}{A}$. It needs the length as well as the area.

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