In part (c) of the problem, you're tasked with determining the resistivity of a material by analyzing the relationship between the quantities of voltage ($\Delta V$), current ($I$), and diameter ($D$) of wires with the same length. The problem asks what quantities could be graphed to yield a straight line for calculating the resistivity.

Part (i): Identifying quantities to graph for a straight line

The resistivity ($\rho$) of a material is related to the resistance ($R$) by the equation:

$R = \rho \frac{L}{A}$

where:

• $L$ is the length of the wire,
• $A$ is the cross-sectional area of the wire, which depends on its diameter $D$, given by $A = \pi \left(\frac{D}{2}\right)^2$.

Also, Ohm’s Law tells us that:

$V = IR$

Rearranging this equation, we get:

$R = \frac{V}{I}$

Now, you can combine these equations to link the quantities that will give a straight line. Substituting $R$ from Ohm’s law into the resistivity equation:

$\frac{V}{I} = \rho \frac{L}{A}$

Rewriting the area $A = \pi \left(\frac{D}{2}\right)^2$:

$\frac{V}{I} = \rho \frac{L}{\pi \left(\frac{D}{2}\right)^2}$

This equation suggests that plotting $\frac{V}{I}$ (which represents resistance) versus $D^{-2}$ (since the diameter is squared in the area formula) should yield a straight line with slope proportional to the resistivity $\rho$.

So, the quantities that should be graphed to get a straight line are:

• Horizontal axis: $D^{-2}$ (diameter squared),
• Vertical axis: $\frac{V}{I}$ (the ratio of voltage to current, representing the resistance).

Part (ii): Plotting the data and drawing the best fit line

For plotting:

1. You would use the grid provided and plot each data point based on the relationship between $D^{-2}$ and $\frac{V}{I}$.
2. You should scale and label the axes clearly with units (mm for diameter and voltage, A for current).

Would you like me to proceed with plotting this data for you or assist further with any of these steps?