Physics 9702/22 — February/March 2019

Background Concept

SI base units are the fundamental units from which all other (derived) SI units are built. There are 7 SI base units:

• metre ($\text{m}$)
• kilogram ($\text{kg}$)
• second ($\text{s}$)
• ampere ($\text{A}$)
• kelvin ($\text{K}$)
• mole ($\text{mol}$)
• candela ($\text{cd}$)

Understanding the Question

You are told that ampere, metre and second are SI base units, and the question asks for two other SI base units (so any two from the remaining four are acceptable).

Approach

Recall the complete list of SI base units and write down any two that are not already mentioned.

Step-by-Step Reasoning

• The given base units are $\text{A}$, $\text{m}$, and $\text{s}$.
• Other valid SI base units include $\text{kg}$, $\text{K}$, $\text{mol}$, and $\text{cd}$.
• Choose two of these and state them clearly.

Key Takeaways

• Know the 7 SI base units.
• Any derived unit can be expressed in terms of these base units.

Common Mistakes

• Giving a derived unit (e.g. $\text{N}$, $\text{J}$, $\Omega$) instead of a base unit.
• Repeating one of the units already stated in the question.

Things to Be Careful About

• The kilogram is a base unit even though it contains a prefix ("kilo-") in its name.
• Write the correct unit symbols (e.g. $\text{mol}$ not "mole" as a symbol).

Background Concept

A key use of SI base units is checking dimensional consistency (homogeneity) of equations. If an equation is correct, both sides must have the same units.

For unit questions:

1. Rearrange the equation to make the target quantity the subject.
2. Replace each quantity with its SI unit (and then in terms of base units).
3. Simplify by cancelling.

Useful unit facts:

• charge: $\text{C} = \text{A s}$
• force: $\text{N} = \text{kg m s}^{-2}$
• speed: $\text{m s}^{-1}$

Understanding the Question

You are given

$$v = \frac{\mu F}{e}$$

where:

• $v$ is the average drift speed (units $\text{m s}^{-1}$)
• $F$ is force on one electron (units $\text{N}$)
• $e$ is electron charge (units $\text{C}$)
• $\mu$ is a constant

The task is to find the SI base units of $\mu$.

Approach

Rearrange to get $\mu$ in terms of $v$, $e$, and $F$. Then write each of those in base units and simplify.

Step-by-Step Reasoning

Start with

$$v = \frac{\mu F}{e}$$

Make $\mu$ the subject:

$$\mu = \frac{ve}{F}$$

Now substitute units:

• Drift speed $v$ has units $\text{m s}^{-1}$.
• Charge $e$ has units coulomb, and

$$\text{C} = \text{A s}$$

• Force $F$ has units newton, and

$$\text{N} = \text{kg m s}^{-2}$$

So

$$[\mu] = \frac{(\text{m s}^{-1})(\text{A s})}{\text{kg m s}^{-2}}$$

Simplify:

• The $\text{m}$ cancels top and bottom.
• For seconds: numerator has $\text{s}^{-1} \times \text{s} = \text{s}^0$ (so none left), and dividing by $\text{s}^{-2}$ is equivalent to multiplying by $\text{s}^2$.

Hence

$$[\mu] = \frac{\text{A s}^2}{\text{kg}} = \text{A s}^2\,\text{kg}^{-1}$$

Key Takeaways

• Always express derived units like $\text{C}$ and $\text{N}$ in SI base units.
• Rearranging first avoids confusion about where quantities sit (numerator/denominator).
• Unit simplification is just algebra with indices.

Common Mistakes

• Using $\text{C}$ as if it were a base unit and not converting to $\text{A s}$.
• Using $F$ in $\text{kg m s}^{-1}$ (that would be momentum, not force).
• Losing track of negative powers of seconds when simplifying.

Things to Be Careful About

• The question asks for SI base units, so the final expression should only involve $\text{kg}$, $\text{m}$, $\text{s}$, $\text{A}$ (and possibly others if they appear, but here only these survive).
• Cancel units systematically; writing indices explicitly (e.g. $\text{s}^{-2}$) helps prevent sign errors.