Question

evaluation and review questions:

1. the slope of a line is the change in the y direction divided by the change in the x direction. the definition for slope is illustrated in figure 4-4. find the slope for each resistor on plot 4-1. note that the slope for a resistor has units for conductance, the siemens.

figure 4-4: a graph with labels \change in y\, \change in x\, and the formula ( \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{2}{3} )

Explanation:

To find the slope (conductance) of a resistor, we use the formula for slope: \( \text{Slope} = \frac{\text{Change in } y}{\text{Change in } x} \).

Step 1: Identify Change in \( y \) and Change in \( x \)

From the graph (Figure 4-4), assume the change in \( y \) (e.g., current, \( I \)) and change in \( x \) (e.g., voltage, \( V \)) are given by the rise and run of the line. For the example in the figure, \( \text{Change in } y = 2 \) and \( \text{Change in } x = 3 \) (or proportional values from the plot).

Step 2: Calculate the Slope

Using the formula:
\[
\text{Slope} = \frac{\text{Change in } y}{\text{Change in } x} = \frac{2}{3} \, \text{siemens} \, (\text{or as per the plot’s scale}).
\]

For a specific resistor on Plot 4-1, follow these steps:

1. Locate two points \((x_1, y_1)\) and \((x_2, y_2)\) on the resistor’s line.
2. Compute \( \Delta y = y_2 - y_1 \) (change in \( y \)) and \( \Delta x = x_2 - x_1 \) (change in \( x \)).
3. Substitute into \( \text{Slope} = \frac{\Delta y}{\Delta x} \) to find conductance (in siemens).

If we use the example from Figure 4-4 (where \( \Delta y = 2 \), \( \Delta x = 3 \)):

Answer:

\( \frac{2}{3} \) siemens (or the value derived from Plot 4-1’s data).