## Analyzing the Given Data and Completing the Table
We are provided with the following data:
| Voltage (V) | 4 | 8 | 12 | 16 | 20 |
| ----------- | -- | -- | -- | -- | -- |
| Current (I) | 0.4 | 0.8 | 1.2 | a | 2.0 |
### (a) Completing the Table
To complete the table, we need to determine the missing value for the current when the voltage is [tex]\(16\)[/tex] V. From the given data, we can infer that there may be a linear relationship between Voltage (V) and Current (I). We will assume Ohm's Law, which is stated as:
[tex]\[ V = I \cdot R \][/tex]
Where:
- [tex]\( V \)[/tex] is the voltage
- [tex]\( I \)[/tex] is the current
- [tex]\( R \)[/tex] is the resistance
Given that the resistance [tex]\( R \)[/tex] is constant, we can find [tex]\( R \)[/tex] using any of the given data points.
Using the first set of values provided ( [tex]\( V = 4 \)[/tex] and [tex]\( I = 0.4 \)[/tex] ):
[tex]\[ R = \frac{V}{I} = \frac{4 \, \text{V}}{0.4 \, \text{A}} = 10 \, \text{ohms} \][/tex]
Now we can use this resistance value to find the missing current when [tex]\( V = 16 \)[/tex]:
[tex]\[ I = \frac{V}{R} = \frac{16 \, \text{V}}{10 \, \text{ohms}} = 1.6 \, \text{A} \][/tex]
Thus, the completed table is:
| Voltage (V) | 4 | 8 | 12 | 16 | 20 |
| ----------- | -- | -- | -- | -- | -- |
| Current (I) | 0.4 | 0.8 | 1.2 | 1.6 | 2.0 |
### (b) Showing the Relationship Between V and I
We must show that the relationship between Voltage (V) and Current (I) is linear and follows the form [tex]\( I = \frac{V}{R} \)[/tex].
### Step-by-Step Verification:
1. Calculate the resistance for each data point pair and confirm they are equal:
- For [tex]\( V = 4 \)[/tex]: [tex]\( R = \frac{4}{0.4} = 10 \, \text{ohms} \)[/tex]
- For [tex]\( V = 8 \)[/tex]: [tex]\( R = \frac{8}{0.8} = 10 \, \text{ohms} \)[/tex]
- For [tex]\( V = 12 \)[/tex]: [tex]\( R = \frac{12}{1.2} = 10 \, \text{ohms} \)[/tex]
- For [tex]\( V = 16 \)[/tex]: [tex]\( R = \frac{16}{1.6} = 10 \, \text{ohms} \)[/tex]
- For [tex]\( V = 20 \)[/tex]: [tex]\( R = \frac{20}{2.0} = 10 \, \text{ohms} \)[/tex]
Since the resistance [tex]\( R \)[/tex] is constant (10 ohms) for all the data points, it confirms that there is a consistent linear relationship between Voltage (V) and Current (I).
### Conclusion
The completed table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Voltage (V)} & 4 & 8 & 12 & 16 & 20 \\
\hline
\text{Current (I)} & 0.4 & 0.8 & 1.2 & 1.6 & 2.0 \\
\hline
\end{array}
\][/tex]
The relationship between Voltage (V) and Current (I) can be expressed as:
[tex]\[ I = \frac{V}{R} \][/tex]
Where [tex]\( R \)[/tex] is the constant resistance, which in this case is [tex]\( 10 \)[/tex] ohms. This demonstrates a direct proportionality between Voltage and Current, consistent with Ohm's Law.
